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Thread: Subsets of R

  1. #1 Subsets of R 
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    Yeah, well, that's all the title bar allows me, but it's not quite what I meant.

    So. I was thinking - just thinking, mind - of extending my (arguably) crappy tutorial on topological spaces to some elementary manifold theory.

    So if such a thread were to make any sense, I would have to be sure of the following, which I want to check on:

    Considered as a vector space, it is certainly true that any proper (or improper) subspace of is isomorphic to itself.

    Q: Is this the only vector space with this property? I think so, but I'm not certain.....

    It is also certainly true that, considered as a topological space, any subspace of is homeomorphic to itself.

    Q: Is this the only topological space with this property? I think so, but I'm not certain.....

    Now consider as a manifold.

    Q: Is it true that any submanifold of is diffeomorphic to itself? This seems less likely, but if so, would this be the only such manifold with this property?


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  3. #2 Re: Subsets of R 
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    Quote Originally Posted by Guitarist
    Yeah, well, that's all the title bar allows me, but it's not quite what I meant.

    So. I was thinking - just thinking, mind - of extending my (arguably) crappy tutorial on topological spaces to some elementary manifold theory.

    So if such a thread were to make any sense, I would have to be sure of the following, which I want to check on:

    Considered as a vector space, it is certainly true that any proper (or improper) subspace of is isomorphic to itself.

    Q: Is this the only vector space with this property? I think so, but I'm not certain.....

    It is also certainly true that, considered as a topological space, any subspace of is homeomorphic to itself.

    Q: Is this the only topological space with this property? I think so, but I'm not certain.....

    Now consider as a manifold.

    Q: Is it true that any submanifold of is diffeomorphic to itself? This seems less likely, but if so, would this be the only such manifold with this property?
    It is NOT true that any proper subspace of is isomorphic to . In fact finite dimensional real vector spaces are classified up to isomorphism by dimension. So no proper subspace will be isomorphic to .

    It is also certainly NOT true that, considered as a topological space, any subspace of is homeomorphic to itself. In fact and are not homeomorphic unless n=m. Moreover there are lots of topological subspaces of that are distinct from . The circle in the plane is one, the 2-sphere in is another.

    When you go to diffeomorphisms the situation is even muddier. There are exotic differentiable structures that crop up. There are exotic structures on that are not diffeomorphic to the usual for instance. Milnor a long time ago showed the existence of several exotic differentiable structures on the 7-sphere. http://en.wikipedia.org/wiki/Exotic_R4


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  4. #3  
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    Yes, you are right. Dunno what came over me, sorry about that.
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  5. #4  
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    does have the nice property that every basic open subset is homeomorphic to the entire real line. Is that what you were thinking of guitarist?
    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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  6. #5  
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    Quote Originally Posted by river_rat
    does have the nice property that every basic open subset is homeomorphic to the entire real line. Is that what you were thinking of guitarist?
    That is only true if n = 1.

    In the case of n>1 what is true is that open balls are homeomorphic to .
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  7. #6  
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    lol, oops let me restate that

    does have the nice property that every basic open subset is homeomorphic to the .
    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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  8. #7  
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    On subsets of we could also have a thread about strange constructions in . Like functions satisfying the strong Darboux prop
    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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  9. #8  
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    Quote Originally Posted by river_rat
    On subsets of we could also have a thread about strange constructions in .
    Start one, start one!
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  10. #9  
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    Lol, when time permits again Jane Bennet I've got to a. recreate the construction or b. get it from a book first!
    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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  11. #10  
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    Quote Originally Posted by river_rat
    Lol, when time permits again Jane Bennet I've got to a. recreate the construction or b. get it from a book first!
    What is the strong Darboux property? I assume that it has something to do with a function on the line assuming intermediate values, but have never run across that particular term.
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  12. #11  
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    A real valued function on the real line is strongly darboux if the image of every open interval is the entire real line.
    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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  13. #12  
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    Quote Originally Posted by river_rat
    A real valued function on the real line is strongly darboux if the image of every open interval is the entire real line.
    That sounds like an interesting construction. Such a function ought to be, in some sense, about as discontinuous as one can possibly get.
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  14. #13  
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    Ah, i remember the construction - everyone here fine with transfinite induction?
    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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  15. #14  
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    Quote Originally Posted by river_rat
    everyone here fine with transfinite induction?
    Iím not.
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