1. As promised. First recall I and others have said that a manifold is a topological space with certain additional properties, and also that a topological space space is a set with certain additional properties. So it seems wise to start with some exceptionally basic set theory.

In what follows, I will cover no more ground than is needed in order to move on. And, oh; if you think you know this stuff already (and I'm sure you do!) it still might be worth a quick skim, as not everyone uses the same notational and terminological conventions.

So a set is simply a collection of objects that has some claim to being a collection. If this sounds circular, it is because the notion of a "set" is in some sense primitive - we all know what we mean, just that we can't quite put our finger on it. I write to mean that is an element in the set and its negation .

Context sometimes make it more convenient to write to mean exactly the same thing - is a set that contains the element , likewise the negation.

Now notice this key fact: elements in a set can be absolutely anything -they may, and often are, themselves sets, as we shall see. So in order to avoid a famous paradox due to B. Russell, I make the stipulation.....

A set is well-defined iff every object in the universe of objects is either definitely in that set or definitely not in it.

Key fact 2: Given a set and another set I will say that if every element in is also an element in I will say say that is a subset of . But note I do not NECESSARILY insist there are elements in that are NOT also in .....but I might.

In the latter case I write , in the former I write Note this does not mean I am uncertain whether or not , rather that whatever I have to say about this subset applies equally whether equality holds or whether it doesn't.

Let me know introduce you some extremely important subsets. Suppose a set. Then the set is called the "singleton set", since it has only a single element. We shall see a generalization of this shortly, but note for now that , since is always true, is meaningless.

The other subset of key importance is the empty set which I write as for any set . In a while I shall prove the rather startling fact that the empty set is always a subset of any set.

This post is already over-long, but let me just mention 2 "operations" on sets (of course they are not really operations)

The intersection of 2 sets defines those elements that are in both sets equally. So for the sets I write . Note that the intersection is itself a set.

The union of 2 sets defines those elements that are in one, the other, or both. I write for the union, which is itself a set.

Some properties of intersection and union will be of interest, but I have no doubt you have had enough of me for now.

2.

3. So specifically refers to proper subset?
Do we get some exercises?

Oh, and thank you very much!

4. Possibly very stupid question going your way. I know no advanced math, so please don't judge...

Originally Posted by Guitarist
A set is well-defined iff every object in the universe of objects is either definitely in that set or definitely not in it.
So what is "the universe of objects"? I'm familiar with the concept from basic work with sets, but in this context, what defines it? Is this the same thing as "u"?

5. Hi! This will be fun.
Originally Posted by Guitarist
So a set is simply a collection of objects that has some claim to being a collection. If this sounds circular, it is because the notion of a "set" is in some sense primitive - we all know what we mean, just that we can't quite put our finger on it.
So you dont use any precaution like demanding the set satisfy the laws of logic!?
I write to mean that is an element in the set and its negation .

Context sometimes make it more convenient to write to mean exactly the same thing - is a set that contains the element , likewise the negation.

Now notice this key fact: elements in a set can be absolutely anything -they may, and often are, themselves sets, as we shall see. So in order to avoid a famous paradox due to B. Russell, I make the stipulation.....

A set is well-defined iff every object in the universe of objects is either definitely in that set or definitely not in it.
Theres the protection then
Can the universe of objects be a set?

6. Originally Posted by sigurdW
Can the universe of objects be a set?
I think that's called the universal set.

7. Originally Posted by guymillion
Originally Posted by sigurdW
Can the universe of objects be a set?
I think that's called the universal set.
In a religious setting somebody claimed there must be a cause of all causes if I claim that everything has a cause. But no, he must prove that there really is a cause for all those causes that dont cause themselves before i believe him. If the universe of objects is a universal set it might contain all sets that do not contain themselves. Guitarist have as defence the principle:"A set is well-defined iff every object in the universe of objects is either definitely in that set or definitely not in it." At first sight it seems sufficient, but it might depend on the definite meaning of the word "definite"... What do you think?

8. Originally Posted by sigurdW
If the universe of objects is a universal set it might contain all sets that do not contain themselves.
That doesn't seem logical. A bit too philosophical for me.

Originally Posted by sigurdW
Guitarist have as defence the principle:"A set is well-defined iff every object in the universe of objects is either definitely in that set or definitely not in it." At first sight it seems sufficient, but it might depend on the definite meaning of the word "definite"... What do you think?
I don't think that there is any question here. I think it is sort of a black and white scenario. However, that makes me wonder: What is a ill-defined set?

• Guitarist was very open when he said he was just mentioning the bare basics of set theory you will need
• That generally implies naive set theory
• We all know naive set theory is inconsistent
• But who cares, we are talking about manifolds and not the schema of specification.
That is all

10. Originally Posted by river_rat
• Guitarist was very open when he said he was just mentioning the bare basics of set theory you will need
• That generally implies naive set theory
• We all know naive set theory is inconsistent
• But who cares, we are talking about manifolds and not the schema of specification.
That is all
Im not so sure naive set theory necessarily is inconsistent.
If you add the requirement that a set exists if and only if it conforms to the laws of logic, then its not obvious that inconsistent set exists.
And ...yes... I understand what this thread is going to be about, but at the moment its about the foundation of set theory isnt it? Guitarist explicitly asked for comments, should I keep quiet?

Im more interested in its conceptual foundation than in using the theory of manifolds for practical purposes.
I expect this thread soon will be way above my (and most laymens) head so I try to examine the foundational concepts as carefully as possible as early as possible. So to sum up my position: When reading I often come across the concept "manifold", and Im happy for a chance to improve my (intuitive) understanding of the concept .

BTW: Please treat us posters in the thread as ignorants in principle and define the concepts in your arguments whenever possible: What is "the schema of specification"?

11. Originally Posted by guymillion
Originally Posted by sigurdW
If the universe of objects is a universal set it might contain all sets that do not contain themselves.
That doesn't seem logical. A bit too philosophical for me.

Originally Posted by sigurdW
Guitarist have as defence the principle:"A set is well-defined iff every object in the universe of objects is either definitely in that set or definitely not in it." At first sight it seems sufficient, but it might depend on the definite meaning of the word "definite"... What do you think?
I don't think that there is any question here. I think it is sort of a black and white scenario. However, that makes me wonder: What is a ill-defined set?
The first such set has been mentioned, its quite famous:the Russell Set.
The set of all sets not containing themselves as an element.

It is easily shown to generate a paradox, the easiest solution is to define it as something else than a set!
"Class" is customarily used, then there can be no class that contains all classes that do not contain themselves...and the traditional defence is to use (and be happy with) the well known "type theoretical" approach.

This seems like nitpicking but it is IMPORTANT that a theory has a solid foundation.
I cant believe guitarist will build his theory of manifolds on clay.

12.

13. Yes indeed!
An unusally competent book: Its strongly recommended as a first reading on the subject!

14. Yeah well, you need to be a bit careful using terms like "universal set" - you run the risk of Russell's paradox. The universe of objects I referred to is really just that - every possible object, physical or otherwise, in the entire universe.

To answer GE, yes, denotes a proper subset. I'll show you more in mo

Let's move on to show that any non-empty set has at least 2 subsets, itself and the empty set .

Suppose is a subset of without for now specifying whether we mean a proper subset or otherwise. Then the rules for set union and set interection must give us that

where I am invoking the rule that an element can appear once and once only in any set (actually there are exceptions, but let's leave that for now)

If I take the above as a definition of a subset, and noting my comment following it, we must also have that

and so by our definition is a subset of itself.

Similarly and so the empty set is also a subset of .

One more thing before moving on: if one says these 2 sets are disjoint meaning, as I trust is obvious, share no elements in common. BUT...even in this case still makes sense, and one calls this the disjoint union. Much much later we shall have a need of a special case of this.

One final thing which is CRUCIAL to what follows. Suppose as before that is subset of without specifying whether or not it is proper. Then the elements in that are NOT in is called the complement of in .

We have two ways of writing this. We may say that the complement is a subset of . More verbosally, we may say that defines the complement. If you look at my LaTex code you will see that this is just the set-theoretic version of the arithmetic "minus".

In fact, using this we can now define the proper subset

Obviously just as in arithmetic. Suppose once again is an unspecified subset of , then if one says that is a proper subset and writes, as before

I think that's about done with the boring stuff, but we next need set functions in some detail. They are a tad more interesting

15. Hi! There will be functions!?
Originally Posted by Guitarist
Yeah well, you need to be a bit careful using terms like "universal set" - you run the risk of Russell's paradox. The universe of objects I referred to is really just that - every possible object, physical or otherwise, in the entire universe.
Must we limit ourselves to the universe?
Possibly there are "Multiuniverses" and "Idunnowhats" out there?

Why not use "Existence"?

Would that not be a guarantee that "Nothing is left out.":

"every possible object, physical or otherwise, in the entire existence"

I do accept all definitions so far since I want to see what comes out of them,
I just cant resist trying to help out a little
Tell me to shut up if Im annoying.

16. Yeah well, Russell's paradox refers to the "universal set of all sets". Let's call it as .

We can easily dispose of this using the notation and concepts now at our disposal.

Consider the complement . Then if is the set of ALL sets, then we must have that (this notation is kosher, as has sets as elements).

But NO set can contain its own complement as a subset (or element as in this case). So there must exist a "super-universal" set, call it , which contains both and . But then again, if is truly universal, then it must contain its own complement, and so the ludicrous chain continues.

There can be no set of all sets.

17. Originally Posted by Guitarist
Yeah well, Russell's paradox refers to the "universal set of all sets". Let's call it as .

We can easily dispose of this using the notation and concepts now at our disposal.

Consider the complement . Then if is the set of ALL sets, then we must have that (this notation is kosher, as has sets as elements).

But NO set can contain its own complement as a subset (or element as in this case). So there must exist a "super-universal" set, call it , which contains both and . But then again, if is truly universal, then it must contain its own complement, and so the ludicrous chain continues.

There can be no set of all sets.
Many years ago I created several web pages on mathematical physics. The main menu is here
Mathematical Physics

The one for manifolds is right here
Introduction to Manifolds

I tried my best to make it as clear and clean as possible. All comments welcome.

18. @sigurdW; You could always start another thread about foundations of set theory and Russels paradox. As long as it doesn't devolve into lunacy(pedantry is okay, maybe even necessary) I would probably participate. As for this thread I would like to get on to the manifolds.

@Guitarist; Thanks! Okay, I've got this so far.

19. GiantEvil,
If youre in a hurry you can now check out what pmb says abut it.
I dont care very much about threads its the people the thread contains I find of interest.
Whats the use of a thread without intelligent visitors?
I do have some results on paradoxes that need checking by peers but where are they?

Definition of Manifold: Let Rn, represent the set of all n-tuples of real numbers (x1, x2, x3 , … , xn). A manifold is a set of ‘points’ M for which each point ofM has an open neighborhood that has a continuous 1-to-1 map onto an open set of Rn for some n. This simply means that M is locally ‘like’ Rn,. The dimension of the manifold is, obviously, n. It is important that the definition involves only open sets and not the whole of M and Rn, because we do not want to restrict the global topology of M. The map is only required to be 1-to-1 and not to preserve lengths or angles. Length is not defined at this level of geometry. In some physical applications the notion of the distance between two points is not required.
By definition, a map associates a point P of M an n-tuple (x1(P), x2(P), x3(P), … , xn(P)). The numbers x1(P), x2(P), x3(P), … , xn(P) are called thecoordinates of P under this map as illustrated in Fig. 1. below

The superscripts do not indicate powers of x(P), they indicate an index to refer to a particular variable. One way of thinking about a manifold is that it is simply any set of points which can be given n independent coordinates in some neighborhood of any point, since these coordinates actually define the required map to Rn.
At this point the reader should have a general idea of what a manifold is. To understand more we must understand the nature of these coordinate maps. Let f be a 1-to-1 map from a neighborhood U of a point P of M onto an open set f(U) of Rn as illustrated in Fig 1. The neighborhood need not contain all of M so there will be neighborhoods with there own maps and each point of M must lie in at least one such neighborhood. The pair consisting of a neighborhood U and its map f is called achart and denoted (U, f). It is easy to see that these open neighborhoods must have overlaps if all points of M are to be included in at least one, and it is therefore overlaps which enable us to give further characterization of the manifold. See Fig. 2 below

As shown in the diagram we have chosen two overlapping neighborhoods, U and V. As shown in Fig. 2 the neighborhood U is mapped into Rn by f while V is mapped into Rn by g. U Ç V is open and is given two different coordinate systems by the two different maps. There is therefore an equation relating the two coordinate systems. To find it, pick a point in the image of U Ç V under f (i.e. a point in Rn). Refer to this point as (x1, x2, x3, … , xn). As shown in Fig. 3 below

Since f is 1-to-1 then it has an inverse map f--1, so there is a unique point in U Ç V which has these coordinates under f. Now let g take us from S to another point in Rn, say (y1, y2, y3, … , yn). We have thus constructed the function g(f--1) which is expressed as

which is called a coordinate transformation. If the partial derivatives of order k or less of all these functions {yi} with respect to the {xi} exist and are continuous, then the maps f and g are said to be Ck related. It is possible to construct a whole system of charts, called an atlas, in such a way that every point in M is in at least one neighborhood and every chart is Ck related to every other one it over laps with, then the manifold is said to be a Ck manifold. A manifold of class C1 is said to be a differential manifold.

20. Personally, I think I need Guitarists more step-by-step approach. Pmb's page dives in too quickly for me!

21. Originally Posted by Strange
Personally, I think I need Guitarists more step-by-step approach. Pmb's page dives in too quickly for me!
A good point but I always find it best to use independent sources when I study something.
A lot can be learned by comparing approaches.

22. sigurdW, nobody seems to be in any particular hurry - in fact I should prefer if they were not, as we have a fair bit of ground to cover before we reach our quarry.If you don't want to take it piecemeal, that is your choice, but I suggest if you DO make that choice, your understanding of manifolds will of necessity be superficial

23. Um, another likely stupid question if you don't mind... Does A(union)B=A(intersection)B imply that A=B?

24. Originally Posted by Guitarist
sigurdW, nobody seems to be in any particular hurry - in fact I should prefer if they were not, as we have a fair bit of ground to cover before we reach our quarry.If you don't want to take it piecemeal, that is your choice, but I suggest if you DO make that choice, your understanding of manifolds will of necessity be superficial
Hey! Its exactly my step by step approach that irritates my thread mates. I wonder what gave you the idea that I want to skip any link in a logical chain? On the contrary... For example: I suggested (mathemathical) existence should be included somewhere at foundational level. Maybe substituting for the "universe of objects". I understand its a complicated matter so I really did not expect an immediate reaction. Meanwhile I wonder if the statement: "There is no sets of all sets." is equivalent to "There is no set of all sets within the universe of objects"? Also I felt I could (for the time being) skip checking the details in your proof of there being no set of all sets but if you really think we should attend to detail, then fill in any missing steps so were sure that everything necessary is there.

25. Originally Posted by epidecus
Um, another likely stupid question if you don't mind... Does A(union)B=A(intersection)B imply that A=B?
(Yes.) But my answer is not a proof. (Why trust anybody?) You should prove it by your self by applying the definitions given.(Probably you did that but didnt trust your ability?)
Its generally not the answer that is important, its how it is reached that is the best part of the story.

26. Originally Posted by epidecus
Um, another likely stupid question if you don't mind... Does A(union)B=A(intersection)B imply that A=B?
Yes
It is also the case that if and then

In case anyone is interested, the second sticky in the maths forum has a Tex tutorial.

27. Originally Posted by GiantEvil
In case anyone is interested, the second sticky in the maths forum has a Tex tutorial.
I was going to say, "I hadn't seen that". But I have already posted in it....

28. Originally Posted by Strange
Personally, I think I need Guitarists more step-by-step approach. Pmb's page dives in too quickly for me!
That was one page in a sequence. You missed the page that followed on set theory. I.e.
Set Theory

29. [QUOTE=pmb;344842]
Originally Posted by Strange
That was one page in a sequence. You missed the page that followed on set theory. I.e.
Set Theory
I have to say, that didn't help much. If you would like some more detailed feedback on what I found hard to follow and why, feel free to start another thread (I don't want to derail this one any more than has already happened).

30. OK, so set functions will be of great interest in what is to follow.

Basically a set function is a gadget that maps elements in one set onto elements in another set (though these sets need not be different from each other). One writes, for the sets that . Read this as "X maps to Y".

I warn you in what follows the conventional notation is a bit of a mess, but it is conventional so we shall use it.

First, if one calls the domain of this function, and its codomain (some people call it the "range" but I don't like that so well).

If and one calls the image of in , and we are free (given certain restrictions that I explain below) to set . Note that somepeople (dare I say usually physicists) call a function - this is wrong, it is an image point, an element in the codomain.

Now here's where the mess comes in. Suppose as before that and that . The one syas that is the pre-image of in . Notice that in this case is NOT a function of any sort (but as you will see, we can make it one in certain cases) rather is (very often) a subset of

Mmm. Well, maybe this will become clearer in a bit.......

One distinguishes 3 distinct types of function, which I define - but note I am choosing my words exptremely carefully. Take them titerally and no more - in other words "read my lips, not my mind"

I continue to use the function(s)

1. If, for all there is at most one such that one says this function is an injection (or injective function);

2. If, for all there is at least one such that one calls this a surjection;

3. Combining these we easily see the following: if, for all there at most one and at least one such that , one calls this a bijection or a one-to-one correspondence

Under these circumstances, we see that, for the surjection, is the set, say . whereas for the injection we can always choose such that

Now to the nightmare. In the case of a bijection, we have that every element in the domains maps to a unique element in the codomain, so that we may assume there is an inverse function, which is annoyingly written as .

I know this is hard to follow, but I strongly recommend you re-read it until it makes sense.

31. Well, a couple of hours and dozens of reads later;

Originally Posted by Guitarist
1. If, for all there is at most one such that one says this function is an injection (or injective function);
From this does it follow that |X|<|Y|?

Originally Posted by Guitarist
2. If, for all there is at least one such that one calls this a surjection;
From this does it follow that |X|>|Y|?

Originally Posted by Guitarist
3. Combining these we easily see the following: if, for all there at most one and at least one such that , one calls this a bijection or a one-to-one correspondence
From this does it follow that |X|=|Y|?

I realize that the cardinality of the sets in question may be a trivial consideration, but I wanted to produce a simple paraphrase to test my understanding of the concepts presented.

32. Originally Posted by GiantEvil
Well, a couple of hours and dozens of reads later;
Oh dear, so sorry about that

I realize that the cardinality of the sets in question may be a trivial consideration, but I wanted to produce a simple paraphrase to test my understanding of the concepts presented.
Yes well, it's not so much that it is trivial, rather it is irrelevant.

It is perfectly true that if there is a bijection from one set to another, then they must have the same cardinality - in fact this is the usual proof of equal cardinality. For the rest I am not sure.

Look - I realize my explanation was rather abstract, and I had (or so I thought) a good reason for that.

Lemme try and be a bit more concrete.....

For our injection we will have that SOME is the image of only one That is, no two elements in can map to the same element in . BUT, there will be elements in that are NOT mapped onto by any element in

For our surjection, EVERY element in is mapped onto by some elements(s) in , and there may well be several that map to a SINGLE element in .

I believe my original was more concise, but I accept it may have been a little hard to follow

33. @Guitarist; No need to apologize, this is stuff I want to learn and there is pleasure in the effort.
Irrelevant is the word I would have used if I had been thinking about it. The word "trivial" has a specific mathematical interpretation that I don't understand yet.
Don't hold back on the abstraction, It's taken awhile, but I'm beginning to understand how it works and the power of it.

I tried to collate all your excellent explanations into one cheat sheet, but I apparently ran into a preset image limit. I'm going to cheat and multi-post just this once.

1. If, for all
there is at most one such that one says this function is an injection (or injective function);
*for the injection we can always choose such that
*For our injection we will have that SOME is the image of only one That is, no two elements in can map to the same element in . BUT, there will be elements in that are NOT mapped onto by any element in X​.

2. If, for all
there is at least one such that one calls this a surjection;
*Under these circumstances, we see that, for the surjection, is the set, say .
*For our surjection, EVERY element in is mapped onto by some elements(s) in , and there may well be several that map to a SINGLE element in .

36. Since now...every thing clear...It is the Set theory

Which relation there is with the manifold...?

Is the manifold an injective, surjective or bijective function?

Can we say that an projection of a 3D object on an 2D plane is an manifold function?... then manifold will be an injective function.

Or the manifold is the 2D set (the projection) obtained from the 3D object?...then a manifold is the set (of n dimension) obtained through an injective function from another set of m dimension (where m>n)?

37. Oh dear, it seems this thread is going nowhere fast

I am sure that GientEvil means well, but if we cannot get past some very basic set theory, then we are in trouble.

As for dapifo, who knows what he is talking about I certainly don't.

And as these have so far been my main interlocutors, I say SHAME on those of you who encouraged me to start this thread.for not contributing

I am about done with it

38. No! I love it! I am reading it! Please! Ahh! Wait!

39. Originally Posted by Guitarist
Oh dear, it seems this thread is going nowhere fast

I am sure that GientEvil means well, but if we cannot get past some very basic set theory, then we are in trouble.

As for dapifo, who knows what he is talking about I certainly don't.

And as these have so far been my main interlocutors, I say SHAME on those of you who encouraged me to start this thread.for not contributing

I am about done with it
I will miss you...I had some questions but decided they could wait.

40. Originally Posted by Guitarist
Oh dear, it seems this thread is going nowhere fast

I am sure that GientEvil means well, but if we cannot get past some very basic set theory, then we are in trouble.

As for dapifo, who knows what he is talking about I certainly don't.

And as these have so far been my main interlocutors, I say SHAME on those of you who encouraged me to start this thread.for not contributing

I am about done with it
Oh well, whatever then, never mind.

EDIT:
Okay, for the sake of guymillion I withdraw my indifference.
But, what about collating the information I've been given implies that I'm unequal to the task of understanding? Huh?
I guess that is what I get for showing up and being interested. I get it now, shut up and nod at the lecturer once in a while. Sheesh!

41. No! Wait, what? Ahh!

I really liked the posts. They helped me learn about set theory, and I'm certain I would stil enjoy them when they get more complicated, even if I couldn't understand it.

Although I understand if you feel as if they are taking up too much time and not getting enough profit.

42. Originally Posted by Guitarist
As for dapifo, who knows what he is talking about I certainly don't.[...]I am about done with it
WelL ... I hope not to be n nuisance for you !!!

I´ll only read ...and see (!?)

43. Originally Posted by GiantEvil
Originally Posted by Guitarist
Oh dear, it seems this thread is going nowhere fast

I am sure that GientEvil means well, but if we cannot get past some very basic set theory, then we are in trouble.

As for dapifo, who knows what he is talking about I certainly don't.

And as these have so far been my main interlocutors, I say SHAME on those of you who encouraged me to start this thread.for not contributing

I am about done with it
Oh well, whatever then, never mind.

EDIT:
Okay, for the sake of guymillion I withdraw my indifference.
But, what about collating the information I've been given implies that I'm unequal to the task of understanding? Huh?
I guess that is what I get for showing up and being interested. I get it now, shut up and nod at the lecturer once in a while. Sheesh!
Hey! Are you sure its not anyone else also annoying Guitarist? Cant it be me? Take it easy.
Dont let your temper prevent you from thinking clearly. I can assure you that to my expert knowledge its a good strategy

44. Originally Posted by dapifo
Originally Posted by Guitarist
As for dapifo, who knows what he is talking about I certainly don't.[...]I am about done with it
WelL ... I hope not to be n nuisance for you !!!

I´ll only read ...and see (!?)
Hi! see my post above. Let us all take it easy.
Im not sure Guitarist is not hot tempered so let us not be so.
He will cool off, he cant help it since he likes mathemathics!
And he correctly thinks we also do.Right?
Encouraging is not equivalent to licking a**.

45. Hi guitarist! Must set theory be the beginning? Couldnt set theory be derived from functions instead?
Just a stray thought: I didnt bother to interrupt with it, but since things are interrupted couldnt you sort of
at least confirm or deny?

46. Hi GiantEvil

To add my two cents again I think the problem here is the lack of threading, so its very easy to get sidetracked into a set theory avenue in a thread about manifolds. Without the ability to create a separate thread may I suggest we create a set theory Q&A topic to address your questions?

Guitarist started quite deep here and there is a chance he will run out of air before getting to the surface of the manifold concept if we take every interesting side branch on the way up.

47. Good idea! Then the threads wont get cluttered up
but we who needs to think things over a couple of times
can do that together in the split off threads!
Any topic really needs undertopics.
It surprises me that its not already built into the system!

48. Originally Posted by Guitarist
Oh dear, it seems this thread is going nowhere fast
In all respect, for what reasons? I'm sure they can be fixed and the thread's purpose can continue swiftly.

I am sure that GientEvil means well, but if we cannot get past some very basic set theory, then we are in trouble.
I'm sorry for contributing myself to this side-track, as it certainly is slowing you down. Respect to your decision; this is only a suggestion: Move the side-track posts to a new thread concerning set theory. Then, the thread will be less cluttered and its purpose would not seem so defeated.

As for dapifo, who knows what he is talking about I certainly don't.
I'm sure his incoherent posts can add quite some frustration, but I hope that fact is not actually part of your choice to abandon this thread.

And as these have so far been my main interlocutors, I say SHAME on those of you who encouraged me to start this thread.for not contributing
In no tone of offense, contributing what? It seems you're not content with the discussion here as it is side-tracking. But what are these quite members supposed to contribute more than what you offer? More questions, I believe... about set theory, as that is what you've been laying out so far. I am sure they are (or better said, were) contently reading this discussion. For example, wallaby, which by the way I'm sure he is genuinely interested in your teaching, said "I'll be watching".

Again, respect to your decision. Just leaving a kind note of reconsideration.

49. well, unless I'm mistaken, I encouraged you to write about this too. I don't see any reason to be ashamed. I am reading your posts but I have nothing to comment on yet. Why not continue? I will be teaching myself linear algebra eventually but I think it would be nice to have an explanation of manifolds to complement my book, which I'm almost finished reading. Pray continue as they said in the old days!

50. Originally Posted by epidecus
For example, wallaby, which by the way I'm sure he is genuinely interested in your teaching, said "I'll be watching".
And indeed i still am.

I (and others) am pretty comfortable with introductory set theory and have been waiting for the "next step" on the road of Manifold Madness. As has been said already, the set theory "side-track" should be shuffled off to a set theory thread. (Although i don't think people who still have questions need to wait for Guitarist to do this)

I definitely appreciate the approach Guitarist is taking (has taken?) here, as opposed to just starting with a definition of Manifolds and going from there. But dealing with side issues, while still forging ahead with the main topic, is just a consequence of this type of format. I hope the show is not over.

51.

52. Gee Guitarist, why are you complaining about my participation in this thread while simultaneously decrying the lack of participation by others?
You did say;
Originally Posted by Guitarist; From post #1
I wasn't the one who went off on a tangent about Russels paradox, in fact I tried to steer the discussion away from that sidetrack...
Originally Posted by GiantEvil; From post#17
@sigurdW; You could always start another thread about foundations of set theory and Russels paradox. As long as it doesn't devolve into lunacy(pedantry is okay, maybe even necessary) I would probably participate. As for this thread I would like to get on to the manifolds.

@Guitarist; Thanks! Okay, I've got this so far.
I even liked the quick explanations given by yourself and river_rat as to the issue. Now I unlike them, HA!

And this seems a little inconsistent;
Originally Posted by Guitarist; Post #21
sigurdW, nobody seems to be in any particular hurry - in fact I should prefer if they were not, as we have a fair bit of ground to cover before we reach our quarry.If you don't want to take it piecemeal, that is your choice, but I suggest if you DO make that choice, your understanding of manifolds will of necessity be superficial
Originally Posted by Guitarist; From post #36
Oh dear, it seems this thread is going nowhere fast
Which is it? What speed do you want to go?

Why have I been singled out here?
Originally Posted by Guitarist; From post #36
I am sure that GientEvil means well, but if we cannot get past some very basic set theory, then we are in trouble.
Didn't I properly answer someone else's simple question on one occasion?
The only part I've hung up on at all is the difference between injective and surjective functions, and all I've done is try to understand. I have not argued about them.

Originally Posted by Guitarist; From post #36
I am about done with it
My temptation is to reply with something hurtful in return, but I would rather learn about manifolds.
Are topological spaces next?

53. Originally Posted by Guitarist
I say SHAME on those of you who encouraged me to start this thread.for not contributing
Don't forget the silent readers - you have done a splendid job so far explaining the basics, and in the light of this I would urge you to remember that many of us will read your posts even if not all of us actively contribute.
I for my part for example have been following this thread since the beginning - I have not posted myself simply because I had nothing relevant and meaningful to contribute. That does not mean however that I haven't learned anything here.

So, in short, the only real shame would be if you let aforementioned "side-trackers" discourage you from completing this thread.

54. Originally Posted by wallaby
I (and others) am pretty comfortable with introductory set theory and have been waiting for the "next step" on the road of Manifold Madness. As has been said already, the set theory "side-track" should be shuffled off to a set theory thread. (Although i don't think people who still have questions need to wait for Guitarist to do this)

I definitely appreciate the approach Guitarist is taking (has taken?) here, as opposed to just starting with a definition of Manifolds and going from there. But dealing with side issues, while still forging ahead with the main topic, is just a consequence of this type of format. I hope the show is not over.
Seconded. I have been able to follow G's "lessons" so far and am also keen to see how this leads on to manifolds (if I don't have to give up before then!).

Might it be worth moving some of the comments into a separate "discussion of" thread?

55. Well I trust I can be forgiven for not realizing so many of you were following this thread. Apologies to all those I offended

OK, let's move on. I assure you all that ALMOST all of what you may have read here is important. The possible exception being the difference between an injection and a surjection. But we need definitely to know what is meant by a bijection. If you want to think of it as a one-to-one correspondence between sets you may - but it is VITAL to realize that a bijection implies an inverse, as I tried to explain.

So to the study of topological spaces, which is called topology. You will notice I use this word in slightly different ways in what follows, but I trust no confusion will arise thereby.

I should also say that in this sense, topology comes in at least two flavours - point-set topology and algebraic topology. The latter is pretty challenging, to me at least, the former far less so. This what we shall be doing here. I should also say I happen to know that river_rat has a postgraduate qualification in topolog and, like all good mathematicians, takes no prisoners. So if he calls me, listen to him

So I start by defining the powerset

Suppose a set (of points). Then the powerset of this set is the set of all subsets of (recall I telegraphed very early on that set elements may themselves be sets). A simple example is worth more than any number of words

Suppose . Then the powerset is .

Now suppose . Then one calls a topology on if and only if the following conditions are satisfied:

1. and are always elements in

2. Arbitrary union of elements in (these are sets recall) are also elements in .

3. Finite intersection of elements in are also elements in (The reason I insist on finite intersections will become clear shortly)

So the ensemble is referred as a Topological Space.

But don't rum away away with the idea that these are two sets somehow "stuck together" - they constitute an indivisible entity, our topological space.

Also note that, for any given set, it is (almost) always possible to define more than one topology.

Clear enough I assume? If not ask questions

56. Originally Posted by Guitarist
Also note that, for any given set, it is (almost) always possible to define more than one topology.
As a neat aside, you can always define at least two different topologies on a given set . The trivial topology and the discrete topology . These will be different unless the set has only one element, which is the only time they coincide

57. I agree with marcus and strange, I've been following along, mainly just waiting until the topic got a little further on.

58. Originally Posted by Guitarist
Well I trust I can be forgiven for not realizing so many of you were following this thread. Apologies to all those I offended

Clear enough I assume? If not ask questions
Ahem... How COULD YOU believe you had no interested audience?! The topic is interesting... the op is interesting...my class mates are interesting...But forget my question, attend instead carefully to your story step by unhurried step like a novel by Agatha Christie

59. Originally Posted by Guitarist
Now suppose . Then one calls a topology on if and only if the following conditions are satisfied ...
Is there an intuitive explanation of why these conditions must apply (or will it become clear later on)?

60. Originally Posted by Strange
Originally Posted by Guitarist
Now suppose . Then one calls a topology on if and only if the following conditions are satisfied ...
Is there an intuitive explanation of why these conditions must apply (or will it become clear later on)?
Arent you thinking in a circle? The conditions define the concept of topology.
What is meant by "topology" is its defining conditions.
Are you asking if the defining conditions are the defining conditions of topology ?

Or asking what intuitive idea or purpose in our minds
makes us select exactly those conditions
for a definition of a new explicit concept?

At first sight I think they were chosen by their consequences.

61. Originally Posted by sigurdW
Arent you thinking in a circle? The conditions define the concept of topology.
What is meant by "topology" is its defining conditions.
Are you asking if the defining conditions are the defining conditions of topology ?
I am asking why a topology is (has to be) defined this way. One could define all sots of arbitrary subsets of the power set, most of them would not be useful. As you put it, I'm asking what are the consequences of a topology having these constraints (being defined in this way).

I assume this definition of topology for sets is related (somehow) to the simple geometric concept (coffee mug = doughnut) of topology. But it isn't clear to me what that relationship is. I assume it formalizes the geometrical relationship somehow. As I say, maybe it will become clear later...

62. Reading around this, it sounds like the definition of a topology in this way allows a homeomorphic function to be defined...

63. Can be {I1}(S) = {S, S,O,O} topological of S?....or there cannot be duplicates elements?

64. So far we have the necessary and sufficient conditions for x being a topology.

Maybe more conditions will come? To define different kinds of topologies?
Then perhaps a certain topology is defined to be...what?

From sets and points we get to topology,
only im not so sure I believe in points,
I mean their individual existence, points in themselves...
Are they convenient fictions, as sets perhaps are?

Edit: Ive temporarily forgotten about functions,and their defining relations to sets.
I mean which defines which: Sets or Functions? An intricate "hen or egg" story?
Id better check if homeomorphic functions are approximations of isomorphic functions.

65. Originally Posted by Strange
Is there an intuitive explanation of why these conditions must apply (or will it become clear later on)?
Hopefully it may, but really it is just a definition. Try this.....

I we accept that every conceivable mathematical structure, say an integral domain, a ring, a field, a monoid, a group, a vector space etc is a set with some "combining" operation, it is usual to insist that the result of "combining" elements of the underlying set remains in that set. One calls this the "closure axiom" for each of these structures.

Now as I said earlier, although set union and set intersection are not strictly speaking operations in that sense, all we are doing is insisting that our topology on , I called it , is closed under union and intersection (subject to the restrictions I gave).

Anyhoo.

Recall our topological space . Elements in (subsets of remember) are called the open sets in .

The set of complements of all elements in are called the closed sets in .

It is a fact that, in topology, sets can be open, closed, both or neither. This can be seen by taking a few simple examples for where , say

If you think this is all a bit airy-fairy, I will shortly show how we can bring it into register with intuition. First, though, I need this....

Suppose , without knowing whether it is open or closed in the above sense. I define the interior of as the union of all open sets contained in . I write for the interior.

Conversely, the closure of is the intersection of all closed sets containing . I write for the closure.

It should be easy to see that if then is open. Conversely, if then is closed.

We can now define the boundary of . Recall, we are not certain yet whether this set is open, closed, both or neither. The boundary of is confusingly written as - this has ABSOLUTELY nothing to do with differentials, it's just a label.

So will have that and . So if then obviously a closed set contains its own boundary as a subset.

Conversely we will have that and , so that if we may say that an open set does not contain its own boundary as a subset.

We can use these factoids to see what it means for set to be both open and closed. Later for that

66. This is important information! :... it is usual to insist that the result of "combining" elements of the underlying set remains in that set. One calls this the "closure axiom" for each of these structures.

This insistance might be because some structures and operations
wont cooperate with the closure axiom...
Therefore its not an axiom in the ancient sense that axioms always are true.

So
dealing with systems we should take care !
Also: The domain or universe to our set theory
(= What we can speak about a la Wittgenstein)
is assumed/assured to be closed?

Also: We have a concept of "nothing" within set theory ...are we sure it captures the essencse of our intuition:

"
One final thing which is CRUCIAL to what follows. Suppose as before that is subset of without specifying whether or not it is proper. Then the elements in that are NOT in is called the complement of in ."

What is or should be the compliment of the empty set? Is really the universal domain in set theory big enough? Is it open or closed for the complement of nothing (= the empty set), is that why there is no set containing everything?

Im NOT saying anything is wrong , Im just checking for eventual problems. And if something is wrong is it really set theory that is to blame? Cant our intuition be wrong instead?

PS The plot thickens, guitarist said exciting things this time I think.

67. Recall that we are talking about topological spaces precisely because we want to get to manifolds.nd if our manifolds are to be "nice" we need toplace some restictions on our spaces. Wait a mo for that - first some notation and then a definition.

I said, and river_rat said, that every set admits of more than one topology. I wrote for a toplogical space. But in practice, one almost never cares what the exact topology is, and simple declares " is a topological space".

But for now we want our to be a certain sort of topological space. First a definition....

Suppose a topological space. Then any open set containing is called a neighbourhood of . Just for now I will refer to the neighbourhood of as .

The first restriction I want to place on is this: if any two distinct points I can always find neighbourhoods such that . Obviously this is a function of the topology, but we can safely say the following:

If the above is true for and there exists some such that, say, then we obliged to say that . This property results from a "separation axiom" catchily called T2, but known to its friends as the Hausdorff property. There are other separation axioms but I will insist on this one.

The other property I shall insist upon is that of being connected. This means pretty much what it seems to mean - our topological space is not comprised of distinct "bits",but let's have definition:

A topological space is said to be connected if and only if t can NOT be written as the union of two disjoint non-empty sets. It is an easy and useful exercise to prove that this is exactly equivalent to saying that the only subsets in that are both open and closed are and . I urge you to try it......

The other preoperty we want is that of compactness. This will take a little whle to explain, but I am running out of battery, so later.

PS sigurdW - if you have nothng substantive to contribute to this (or any other) thread I suggest you stay out. I am well able to ensure that you do, so don't push your luck

68. Hi Guitarist. Just a couple of corrections:

Originally Posted by Guitarist
If the above is true for and there exists some such that, say, then we obliged to say that
Just check your wording here - I'm not sure if what you wrote is what you meant.

Originally Posted by Guitarist
A topological space is said to be connected if and only if t can NOT be written as the union of two disjoint non-empty sets.
You dropped the open set requirement here

69. Originally Posted by Guitarist
Recall that we are talking about topological spaces precisely because we want to get to manifolds.nd if our manifolds are to be "nice" we need toplace some restictions on our spaces. Wait a mo for that - first some notation and then a definition.

I said, and river_rat said, that every set admits of more than one topology. I wrote for a toplogical space. But in practice, one almost never cares what the exact topology is, and simple declares " is a topological space".

But for now we want our to be a certain sort of topological space. First a definition....

Suppose a topological space. Then any open set containing is called a neighbourhood of . Just for now I will refer to the neighbourhood of as .

The first restriction I want to place on is this: if any two distinct points I can always find neighbourhoods such that . Obviously this is a function of the topology, but we can safely say the following:

If the above is true for and there exists some such that, say, then we obliged to say that . This property results from a "separation axiom" catchily called T2, but known to its friends as the Hausdorff property. There are other separation axioms but I will insist on this one.

The other property I shall insist upon is that of being connected. This means pretty much what it seems to mean - our topological space is not comprised of distinct "bits",but let's have definition:

A topological space is said to be connected if and only if t can NOT be written as the union of two disjoint non-empty sets. It is an easy and useful exercise to prove that this is exactly equivalent to saying that the only subsets in that are both open and closed are and . I urge you to try it......

The other preoperty we want is that of compactness. This will take a little whle to explain, but I am running out of battery, so later.

PS sigurdW - if you have nothng substantive to contribute to this (or any other) thread I suggest you stay out. I am well able to ensure that you do, so don't push your luck
You seem to be heading toward compact Hausdorff spaces to make a point.

Is that true?

70. Originally Posted by river_rat
Hi Guitarist. Just a couple of corrections:

Originally Posted by Guitarist
If the above is true for and there exists some such that, say, then we obliged to say that
Just check your wording here - I'm not sure if what you wrote is what you meant.

Originally Posted by Guitarist
A topological space is said to be connected if and only if t can NOT be written as the union of two disjoint non-empty sets.
You dropped the open set requirement here

You dropped the open set requirement here

This does not matter, he is trying to make some other point.

71. Originally Posted by Guitarist
Recall that we are talking about topological spaces precisely because we want to get to manifolds.nd if our manifolds are to be "nice" we need toplace some restictions on our spaces. Wait a mo for that - first some notation and then a definition.

I said, and river_rat said, that every set admits of more than one topology. I wrote for a toplogical space. But in practice, one almost never cares what the exact topology is, and simple declares " is a topological space".

But for now we want our to be a certain sort of topological space. First a definition....

Suppose a topological space. Then any open set containing is called a neighbourhood of . Just for now I will refer to the neighbourhood of as .

The first restriction I want to place on is this: if any two distinct points I can always find neighbourhoods such that . Obviously this is a function of the topology, but we can safely say the following:

If the above is true for and there exists some such that, say, then we obliged to say that . This property results from a "separation axiom" catchily called T2, but known to its friends as the Hausdorff property. There are other separation axioms but I will insist on this one.

The other property I shall insist upon is that of being connected. This means pretty much what it seems to mean - our topological space is not comprised of distinct "bits",but let's have definition:

A topological space is said to be connected if and only if t can NOT be written as the union of two disjoint non-empty sets. It is an easy and useful exercise to prove that this is exactly equivalent to saying that the only subsets in that are both open and closed are and . I urge you to try it......

The other preoperty we want is that of compactness. This will take a little whle to explain, but I am running out of battery, so later.

PS sigurdW - if you have nothng substantive to contribute to this (or any other) thread I suggest you stay out. I am well able to ensure that you do, so don't push your luck

PS sigurdW - if you have nothng substantive to contribute to this (or any other) thread I suggest you stay out. I am well able to ensure that you do, so don't push your luck

Do we have dictators in the area of logic. If you are in command of your reasoning, you will explain your valid logic to the posters, otherwise you are not in command of your logic.

72. When I hear "topological" I think of geometric entities, like a Mobius strip for example. So how is this definition of a topological space related to the intuitive context of topology? Or is that where we're heading to now?

73. Originally Posted by epidecus
When I hear "topological" I think of geometric entities, like a Mobius strip for example. So how is this definition of a topological space related to the intuitive context of topology? Or is that where we're heading to now?
A topological space has certain properties that make it what it is. The geometric entities that may be familiar to us may indeed have properties of topological spaces, but we didn't learn about them in that way.

And I too find this thread interesting because it's not as formal as the way textbooks, of course, have to be written. I have a topology book by Mendelson I got cheap earlier this year and I've been trying to chunk my way through it. And I've actually had at least some limited success!

So, keep it going!

74. Here's a link to a free PDF on topology. It's one of the better free things I've found. It even looks decent on a kindle.

Topology Without Tears; Sidney A. Morris; Sid Morris; Sidney Morris; Shmuel Morris

75. Sorry, I have been little remiss of late, as we have had, and in fact still have, visitors.

Lemme say straight away I concede river_rat's corrections. The second (where I omitted the word "open" in the definition of connectedness) is trivial.

The first (dealing with the Hausdorff property) is more serious, as it could easily lead to confusion. Let me rephrase:

The Hausdorff property of a top.space demands that. for any 2 points , then I can always find neighbourhoods such that .

However, it does NOT follow that for some other point that the SAME neighbourhood of has an empty intersection with any neighbourhood of . This what my clumsy wording seemed to imply.

But Hausdorff insists that there exists SOME neighbourhoods of , say such that provided only that .

OK, let's start on compactness. Note the following is extremely compressed, due to our invasion - try drawing diagrams to help you. First this:

A class of subsets of is called a cover if the union of all is . And if each is open, then it is called an open cover, reasonably enough. Notice that any top.space may admit of more that one cover.

A subclass of for which the union of all is still is called a subcover. And if there be only finitely numbers of these subsets for any cover of whose union is STILL , one says that is compact.

Now hold tight

So a cover for , say the class of subsets is called locally finite if, for all the neighbourhoods say intersect non-trivially with only finitely may (by "non-trivially" I mean the intersection is not empty).

Remember a subcover is a subclass of a cover. So, the refinement of a cover is itself a cover comprised of subsets of all elements (subsets of , recall) in the cover.

Thus a Hausdorff top. space is said to be para-compact if every open cover has a locally finite refinement What does this mean?

Extremely roughly it means something like this. The Hausdorff property does not guarantee that 2 distinct points are not "infinitely far apart" . Para-compactness ensures that they are not

76. I do appreciate your efforts to explain manifolds but perhaps you could cite some examples of why these dry definitions are important by themselves.

Why would anyone bother to construct and describe such entities in the first place? I felt always that mathematics was made unnecessarily obtuse by the choice of such arcane notation and descriptions.

Again nothing personal; I'm looking forward to the meat of your discussion - but sometimes a bit of razzle-dazzle can enliven a topic, you know? Not to mention make it more clearly understood.

77. Originally Posted by ballyhoo
I felt always that mathematics was made unnecessarily obtuse by the choice of such arcane notation and descriptions.
A lot of people feel that way, including myself sometimes. However, one must recognise that mathematical rigour is essential in order to ensure that all the various elements which constitute the overall construct called "mathematics" are internally self-consistent and free of logical contradictions; that is the great beauty and achievement of mathematics.
In practical terms a lot of this detail may seem irrelevant, but I can assure you it isn't.

78. Originally Posted by ballyhoo
why these dry definitions are important by themselves.
It is my earnest hope that all will become clear in due course, although this.....

I felt always that mathematics was made unnecessarily obtuse by the choice of such arcane notation and descriptions.
.....almost get my goat.

1. You are (I assume) a visitor to MathLand. The inhabitants there speak their own language and have their own customs. It is never appropriate to question these, as a visitor to any new land. Is the French language obtuse to you? Do Chinese customs seem arcane to you?;

2. I have done my best to explain the language and customs in MathLand If I failed, that is one thing. If you dislike the language and customs in there, best go home to those you are more familiar with.

Grrrr.

79. While I can appreciate that individual fields by necessity must develop their own 'language' I do believe that elegance and simplicity and expressiveness are important.....besides, do you ever try learning mandarin Chinese or Japanese?! It's like learning stylized cuneiform....unnecessary memorization and complexity which seems designed to make writing obscure! Alphabet for the win! Sometimes a picture says a thousand words such as the first video in my PCA thread. I think more easily expressed and read notation would be nice along with graphical explanations.

80. Originally Posted by ballyhoo
besides, do you ever try learning mandarin Chinese or Japanese?! It's like learning stylized cuneiform....unnecessary memorization and complexity which seems designed to make writing obscure! Alphabet for the win!
You couldn't be more wrong.
Yes, I did try, since I spent quite some time in China on a work contract, and you simply cannot get by over there without learning at least the basics of standardized Mandarin. Due to the very nature of the Chinese language ( same syllables with multiple different meanings ) it would simply not be possible to write it using any form of alphabet, since no one would understand what it is that was written down. Furthermore, there are a large number of regional dialects within China, most of which are so different from one another as to be mutually unintelligible. A common writing system independent of verbal pronounciation is the only way to overcome this, and the motivation is much the same in modern mathematics.
The difficulty of learning written Chinese is, in my mind, often vastly overstated; the average person can get by perfectly well on some 3000 or so characters, most of which are composed of combinations of more basic, repeating elements. Even knowing about half of that number of characters will allow you to roughly understand the content of many everyday texts like newspapers etc. Given enough time and a bit of determination it is not a huge deal to learn this system of writing, as clearly demonstrated by about 1.2 billion people !!

In my opinion the Chinese ideographic writing system is definitely superior to alphabet-based ones, even though it takes more effort to learn.

81. Originally Posted by Markus Hanke
Originally Posted by ballyhoo
besides, do you ever try learning mandarin Chinese or Japanese?! It's like learning stylized cuneiform....unnecessary memorization and complexity which seems designed to make writing obscure! Alphabet for the win!
You couldn't be more wrong.
Yes, I did try, since I spent quite some time in China on a work contract, and you simply cannot get by over there without learning at least the basics of standardized Mandarin. Due to the very nature of the Chinese language ( same syllables with multiple different meanings ) it would simply not be possible to write it using any form of alphabet, since no one would understand what it is that was written down. Furthermore, there are a large number of regional dialects within China, most of which are so different from one another as to be mutually unintelligible. A common writing system independent of verbal pronounciation is the only way to overcome this, and the motivation is much the same in modern mathematics.
The difficulty of learning written Chinese is, in my mind, often vastly overstated; the average person can get by perfectly well on some 3000 or so characters, most of which are composed of combinations of more basic, repeating elements. Even knowing about half of that number of characters will allow you to roughly understand the content of many everyday texts like newspapers etc. Given enough time and a bit of determination it is not a huge deal to learn this system of writing, as clearly demonstrated by about 1.2 billion people !!

In my opinion the Chinese ideographic writing system is definitely superior to alphabet-based ones, even though it takes more effort to learn.

Absolutely not so.

I couldn't be more correct - outside of China and Japan (which borrowed the writing system) it is widely recognized that alphabetic systems are vastly superior. Even other Asian nations citizens seem to agree.

To say that memorization of 3000 some symbols is a reasonable requirement to read and write a written language is absurd. The phonetic alphabet provides far more power to express, read and interpret novel words that any ideographic system could have ever.

Choosing a single language written for governmental affairs in a phonetic alphabet empowers everyone to not only read the language but also communicate verbally and write with speed and minimal investment in time and energy as well as many additional banefits in record-keeping functions such as alphabetic dictionaries for example.

Comparing mathematical notation to ideographic written languages is flawed because such notation has no way of expressing many linguistic structures.

A trivial example would be encountering an unknown ideographic glyph compared to an alphabetic word and trying to communicate that verbally to someone far away who could provide you with a definition. Simplicity itself when spelling with an alphabet - an exercise in dubious artistic description with kanji glyphs etc.

Alphabets are strong in all arenas from ease of writing and reading to ease of speaking unknown words.

p.s.- guitarists we are straying from the path; please continue.

82. it is widely recognized that alphabetic systems are vastly superior.
I disagree.

To say that memorization of 3000 some symbols is a reasonable requirement to read and write a written language is absurd.
1.2 billion people disagree with you !

Choosing a single language written for governmental affairs in a phonetic alphabet empowers everyone to not only read the language but also communicate verbally and write with speed and minimal investment in time and energy as well as many additional banefits in record-keeping functions such as alphabetic dictionaries for example.
Not possible in the case of Chinese though !

Let's face it - this has nothing to do with the topic of this thread, and besides is based solely on personal opinion. There is no need to get into an argument over this.
I am just wondering whether you actually know any Chinese ? If not I don't believe you are qualified to make a judgement on this - you can only give a personal opinion.

83. I know some mandarin Chinese and Japanese also at a basic and intermediate level respectively. I agree we should let it rest and continue with the manifold discussion.

84. Originally Posted by ballyhoo
I felt always that mathematics was made unnecessarily obtuse by the choice of such arcane notation and descriptions.
You should try reading some algebraic proofs from before algebraic notation was developed. Trying to follow a purely verbal description is incredibly torturous. It is hard to believe that so much progress was made in advanced math before these notations were developed.

Guitarist's explanations would be much harder to follow if every relationship had to written out in full. You would get lost in a maze of words. (IANAM - I am not a mathematician. But I can read Japanese!)

85. well, a few pictures or computer generated graphics would help not hinder in my opinion....I have no problem with guitarist continuing to use his notation but I'm simply saying that as far as math as concerned......the more pictures the better......the pca video being a case in point to understand better why and how something happens and is represented.

86. Ballyhoo, guitarist is doing this very much in the Bourbaki style of mathematical prose - and thus applications or pictures do not feature highly.

87. Perhaps off topic a bit.

88. Yes, well, after that little diversion let's now cut to the chase.

Knowing what we now know, we can define MANIFOLD, but first this.....

Suppose top.spaces and the function . Then whenever is an open subset and the pre-image set is an open subset in , one says that this function is continuous. If this is too abstact for you, we can say this more verbosely (but less correctly) as follows

Suppose the same function. Then if is open in (recall we say this is a neighbourhood of the point ) then is continuous at iff is a neighbourhood of such that .

Or to put even less accurately: However "large" might be, a continuous function will "squeeze" the images set down to "fit inside"

This rendition is unsatisfactory because, for a general top. space the concept of "smallness" may have no real meaning.

And if our has an inverse (i.e. is bijective), and the inverse is continuous according to the same criterion, the one calls it a homeomorphism

So I now quickly define a manifold: Excited? You should be if you have read this far!

If, for a topological space , where, say is a neighbourhood of an arbitrary point, the homeomorphism where "lives in" an open subset of , and if this is true for the neighbourhoods of ALL , one calls as a manifold.

Which is no more (or less) than to say that our manifold is locally indiistuishable from some .

All this has been said by me and others, but, admittedly, less boringly

89. Well, that was resolved quicker than I thought.
Thanks Guitarist, you deserve a beer!

90. Is that it? Seems a bit anticlimatic........

91. Originally Posted by ballyhoo
Is that it? Seems a bit anticlimatic........

If you feel sure the last word on the subject is said,
then why dont you ask for a few exercises/questions?

92. Originally Posted by Guitarist
Which is no more (or less) than to say that our manifold is locally indiistuishable from some .
This beautifully encapsulates that entire idea of manifolds. Thanks Guitarist.

93. Originally Posted by ballyhoo
Is that it? Seems a bit anticlimatic........
What exactly where you expecting?

94. Oh good, I was beginning to think I was whistling in the dark. I thank you all for reading and to those that flattered me.

But ballyhoo: I am pleased to say that no, that is NOT it, we are only just starting.

First this. I am about to switch notation, which is considered "frightfully bad form" in polite circles. I offer 2 excuses;

1. I need to free up some symbols (which is no excuse at all!)

2. I "turned" our top.space into a manifold, which from now on I shall call as and a typical point as .

So the definition of a manifold I gave earlier is a quite general one, but I want to restrict it to something that is (relatively) easy to work with. So, given all the boring old shite I said earlier, I will insist that, as a top. space, is connected, Hausdorff, compact and para-compact.

So I suppose that, as before, is a neighbourhood of . And, to recap., the homeomorphism for some integer .

So the pair is called a (local) chart on our manifold. Let us NOT assume that our manifold can be covered by a single (global) chart - this would imply it is "flat" in the sense of Euclid, and therefore pretty boring. Rather let us assume that, for each point in we may have, not only a unique neighbourhood, but also a (possibly) unique homeomorphism

So, for the points I may have the charts , say. The collection of all such charts on is called an atlas, reasonably enough.

But the restrictions I placed on the topology of our manifold dictate that each and every point "lives in" more than one neighbourhood, each with its own homeomorphism to . We shall see the consequence of this shortly, but first this:

Suppose the point , and the homeomorphism . Then obviously is a point here. Now since points in are just the string of numbers (its called an n-tuple), one uses the slightly confusing notation that

I say confusing, though it may not be to you, that this particular n-tuple is an image point in of the point .

So I said above that any point must "live in" at least 2 neighbourhoods, each with their own continuous and invertible map into . Or rather, for any point there are at least 2 relevant charts, let's call them as with image points and .

And if we insist on the "reality" of our point , then clearly there must be some functional relation between these image points.

This requires some notational conventions - but this post is already over-long

95. Originally Posted by Guitarist
If, for a topological space , where, say is a neighbourhood of an arbitrary point, the homeomorphism where "lives in" an open subset of , and if this is true for the neighbourhoods of ALL , one calls as a manifold.
I was following everything (just) until we got here. (It is a bit like listening to a long joke and then not understanding the punchline!)

I think I just need to work at it a bit more (and the next post from G helps).

One question: what determines in ?

96. Originally Posted by Strange

I was following everything (just) until we got here. (It is a bit like listening to a long joke and then not understanding the punchline!)

I think I just need to work at it a bit more (and the next post from G helps).

One question: what determines in ?
Ha ha, funny - for me it was just the opposite. I was struggling a bit until we got to this sentence - now it all becomes clear !

97. Originally Posted by Strange

One question: what determines in ?
Why - the homeomorphism!

Look, I have been asked more than once (I think) to provide concrete examples. I have been reluctant to do this for mine own reasons (which are good ones!).

But try this: Consider the real line . You will have met this girl in school, but almost certainly were not told that is a topological space. But it is - check the definitions. Check also it is connected but not compact, just for fun

Now take a segment (i.e. a "bit") of this real line and join this segment end-to-end. A circle, right?

Check now that, as a top. space our circle is both connected and compact.

But suppose we take the smallest possible arc of our circle and map it back to some little bit of the real line , we can say that this little arc is pretty much the same as some suitably small segment of .

This "pretty much same-ness" I called a homeomorphism, so that our circle is a 1- manifold which we call as or the 1-sphere.

Finally try to show there can be no homeomorphism - no one-to-one correspondence - with any other . I grant you this is less easy, but give it a shot, if only for your own satisfaction

98. Originally Posted by Guitarist
Look, I have been asked more than once (I think) to provide concrete examples. I have been reluctant to do this for mine own reasons (which are good ones!).
I can understand that - it actually makes me work through it more than having examples (or picture ) might.

So can we think of n as the number of dimensions? (Is that all it is, or is it subtler/more complex than that?)

99. Hi Strange

Defining dimension of a manifold properly is a bit technical but I don't think you would go very wrong equating the dimension of the target real space and the dimension of the manifold. The trick is proving that is not homeomorphic to if .

100. Oh hey! Look what the "Similar Threads" tool has dug up;
Manifolds

101. Originally Posted by sigurdW
GiantEvil, If youre in a hurry you can now check out what pmb says abut it. I dont care very much about threads its the people the thread contains I find of interest. Whats the use of a thread without intelligent visitors? I do have some results on paradoxes that need checking by peers but where are they? Im supporting this thread...guess why!? Definition of Manifold: Let Rn, represent the set of all n-tuples of real numbers (x1, x2, x3 , … , xn). A manifold is a set of ‘points’ M for which each point ofM has an open neighborhood that has a continuous 1-to-1 map onto an open set of Rn for some n. This simply means that M is locally ‘like’ Rn,. The dimension of the manifold is, obviously, n. It is important that the definition involves only open sets and not the whole of M and Rn, because we do not want to restrict the global topology of M. The map is only required to be 1-to-1 and not to preserve lengths or angles. Length is not defined at this level of geometry. In some physical applications the notion of the distance between two points is not required. By definition, a map associates a point P of M an n-tuple (x1(P), x2(P), x3(P), … , xn(P)). The numbers x1(P), x2(P), x3(P), … , xn(P) are called thecoordinates of P under this map as illustrated in Fig. 1. below The superscripts do not indicate powers of x(P), they indicate an index to refer to a particular variable. One way of thinking about a manifold is that it is simply any set of points which can be given n independent coordinates in some neighborhood of any point, since these coordinates actually define the required map to Rn. At this point the reader should have a general idea of what a manifold is. To understand more we must understand the nature of these coordinate maps. Let f be a 1-to-1 map from a neighborhood U of a point P of M onto an open set f(U) of Rn as illustrated in Fig 1. The neighborhood need not contain all of M so there will be neighborhoods with there own maps and each point of M must lie in at least one such neighborhood. The pair consisting of a neighborhood U and its map f is called achart and denoted (U, f). It is easy to see that these open neighborhoods must have overlaps if all points of M are to be included in at least one, and it is therefore overlaps which enable us to give further characterization of the manifold. See Fig. 2 below As shown in the diagram we have chosen two overlapping neighborhoods, U and V. As shown in Fig. 2 the neighborhood U is mapped into Rn by f while V is mapped into Rn by g. U Ç V is open and is given two different coordinate systems by the two different maps. There is therefore an equation relating the two coordinate systems. To find it, pick a point in the image of U Ç V under f (i.e. a point in Rn). Refer to this point as (x1, x2, x3, … , xn). As shown in Fig. 3 below Since f is 1-to-1 then it has an inverse map f--1, so there is a unique point in U Ç V which has these coordinates under f. Now let g take us from S to another point in Rn, say (y1, y2, y3, … , yn). We have thus constructed the function g(f--1) which is expressed as which is called a coordinate transformation. If the partial derivatives of order k or less of all these functions {yi} with respect to the {xi} exist and are continuous, then the maps f and g are said to be Ck related. It is possible to construct a whole system of charts, called an atlas, in such a way that every point in M is in at least one neighborhood and every chart is Ck related to every other one it over laps with, then the manifold is said to be a Ck manifold. A manifold of class C1 is said to be a differential manifold.
so do all these pictures capture everything you said till now?

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