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Thread: Motivation for the definition of a topology

  1. #1 Motivation for the definition of a topology 
    Forum Professor river_rat's Avatar
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    In Guitarist's manifolds thread a question was asked about the reasons for defining a topology the way we do. I hope to give a bit of light on this.

    The trick is to realize that the standard definition of a topology was chosen because it was easiest to work with, not because it was most enlightening. So we have to develop an equivalent way of thinking about what we want to talk about in topology first, abstract that idea properly and than show that we can come to the same set of standard definitions.

    So the first question is: What are we trying to study in topology. The answer there is continuity. Now continuity is intricately tied up with the idea of nearness. Recall the first version of the continuity definitions you may have heard. A function on the reals is continuous if nearby points in the domain can't be mapped to far away points in the image. You can extend that to metric spaces with the usual version of continuity but what is important was the idea of nearness. I look at the image and at points that are nearby and than look at the pre-image and points that are nearby and make sure those two sets are related correctly.

    Ok, so we know that the idea of nearness is important - lets abstract it. Lets suppose I have a set and on it I want to have some idea of nearness. By that I mean that given any subset of I can tell you what points are near to . I do this by creating a function which tells me which points are near to a subset (namely the points of are near to the points of ).

    Now lets see what properties this function should have (ie. what are the abstract ideas of nearness we want to capture?).
    • Well should be near to itself for any reasonable definition of nearness. This implies that
    • should find all points that are near to . This means that if we ask what is near to we should not get any new points i.e. .
    • The points which are near to subsets and should be the points near to and the points near to i.e.
    • Finally, nothing should be close to the empty set. ie.
    The second requirement captures the transitive nature of nearness, if x is near to y and y is near to z than x should be near to z. If there was a point then x would be close to some other point which was close to a point in but not close to any point in , breaking this transitive relationship.

    The last requirement is kind of a symmetry argument, as no point in the set should be important a priori. This implies we have two choices for . If we do not choose it to map to the empty set but want to maintain that no point is important we are forced to map it to (else the points we are either including or excluding have some sort of special status). But for all subsets of we have that that so . To keep things interesting we have to make the first choice,

    Now that we have out idea of nearness (a few of you may notice that I have just defined a Kuratowski closure operator), we can give a definition of a continuous function. Recall that we want the image of nearby sets to be nearby. We codify this idea by saying a function from to is continuous if where and are the concepts of nearness on and respectively.

    What does this have to do with the normal definition of a topology, I hear you ask? Well, here is the trick. In this formulation, we call a subset closed if and open if its set compliment is closed. These open sets satisfy the normal definition of a topological space. In fact we can reverse the construction and given a family of subsets which satisfies the normal definition of a topology construct its closure operator. Also, families of subsets tend to be easier to work with than operators (especially when trying to build new topological spaces) and so a topological space is typically defined from the view point of open sets and not from closure operators, even though these view points are equivalent.


    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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    Brassica oleracea Strange's Avatar
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    Thanks for that. I think I just about follow it. I just hope I don't have to answer any questions on it!

    The only bit I am struggling with is this....
    Quote Originally Posted by river_rat View Post
    In this formulation, we call a subset closed if and open if its set compliment is closed. These open sets satisfy the normal definition of a topological space.
    I will have to think a bit more about why it is the open sets that form a topological space. And didn't Guitarist say that there could be both open and closed sets in a topological space...


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    Forum Professor river_rat's Avatar
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    Quote Originally Posted by Strange View Post
    I will have to think a bit more about why it is the open sets that form a topological space. And didn't Guitarist say that there could be both open and closed sets in a topological space...
    Hi Strange, you can define topology just an easily with families of closed sets - all these ideas are complimentary. Every open set A corresponds to a closed one (namely ) and similarly every closed set corresponds to an open one. So we could say that a topological space consists of a family of subsets of which we call closed which are closed under finite unions and arbitrary intersections and where both the empty set and the entire space are closed. What is important is that these sets are enough to fully encode our idea of nearness which is what we wanted in the first place.

    I hope this doesn't add to your confusion but there exist topological spaces where each open set is also closed
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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  5. #4  
    Brassica oleracea Strange's Avatar
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    Quote Originally Posted by river_rat View Post
    [What is important is that these sets are enough to fully encode our idea of nearness which is what we wanted in the first place.
    Yep. That is what I am trying to hang on to here! And I can sort-of see how this relates to the the idea of the topology of things like a coffee cup and a doughnut (and continuous functions mapping between them).

    Likewise, I am just about keeping my nose above water in Guitarist's original thread!
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  6. #5  
    Forum Professor river_rat's Avatar
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    Quote Originally Posted by Strange View Post
    Yep. That is what I am trying to hang on to here! And I can sort-of see how this relates to the the idea of the topology of things like a coffee cup and a doughnut (and continuous functions mapping between them).

    Likewise, I am just about keeping my nose above water in Guitarist's original thread!
    The coffee cup and doughnut idea comes from a concept called Homotopy. Allan Hatcher's book Algebraic Topology Book is a good place to start on that if it interests you.

    But if you need help with anything else just give me a shout
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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