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Thread: Compact Topological Spaces

  1. #1 Compact Topological Spaces 
    Forum Professor wallaby's Avatar
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    So it's taken me a few days to put this together, in that time i think i've come to grasp the definition of a Compact Topological Space (as given by Guitarist). But i've been trying to figure out how we prove that a topological space is compact.

    In my studies of various other sources it's been stated that if a topological space can be mapped onto a closed and bounded subset of then is compact. To support this statement i guess it's necessary to show that; Closed and bounded subsets of are compact and that the image of a compact space is also compact, provided that the mapping is a bijection. But i'm having trouble following some of the various proofs around on the internet.

    That a closed subset of would be compact seems intuitive enough, but i want to understand how we show this in a mathematically rigorous way. The definition of a compact space might come in handy, i'm guessing, so we would be required to show that there is a finite cover of the closed subset in question and that this is a subcover of .


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    Be careful. In the usual topology in R, closed sets are compact only if bounded.


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  4. #3  
    Forum Professor river_rat's Avatar
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    Hi wallaby

    The Heine-Borel theorem shows that for a metric space (i.e. the reals) being compact is the same as being closed and bounded. Its easy to see that you need the bounded part, as the real space its self is closed but definitely not compact.
    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  
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    Actually, I am at a bit of a loss as to why Wallaby thimks that a proof the the Heine-Borel theorem woould help his general understanding of compactness. It is a rather special case applied to .

    For what it's worth - probably not much - H-B, is a specialization of the following theorem:

    The compact subsets of an Hausdorff space are closed.

    The proof is tricky (but not ferocious!), whereas the proof that, when our Hausdorff space is , these closed subsets must also be bounded is much easier.

    I repeat - how does help our more general understanding of compactness?
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  6. #5  
    Forum Professor wallaby's Avatar
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    Quote Originally Posted by Guitarist View Post
    Actually, I am at a bit of a loss as to why Wallaby thimks that a proof the the Heine-Borel theorem woould help his general understanding of compactness. It is a rather special case applied to .
    As i mentioned in the OP i think i understand the definition of compactness, now i'm just following detours. (since i have nothing else to do) I just got finished with the Heine-Borel theorem so i might take a look at the proof of the more general theorem.
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  7. #6  
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    Quote Originally Posted by river_rat View Post
    Hi wallaby

    The Heine-Borel theorem shows that for a metric space (i.e. the reals) being compact is the same as being closed and bounded. Its easy to see that you need the bounded part, as the real space its self is closed but definitely not compact.
    Oops, just saw a typo - the metric space must be complete as well. So subset of a complete metric space compact subset closed and bounded in that metric.
    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  
    Forum Professor river_rat's Avatar
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    Quote Originally Posted by wallaby View Post
    As i mentioned in the OP i think i understand the definition of compactness, now i'm just following detours. (since i have nothing else to do) I just got finished with the Heine-Borel theorem so i might take a look at the proof of the more general theorem.
    As a general rule, proving compactness is actually quite difficult and technical - looking at the conditions for a function space to be compact for example. In fact just proving that the product of compact spaces is compact is non-trivial.
    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  
    Forum Professor wallaby's Avatar
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    Quote Originally Posted by river_rat View Post
    As a general rule, proving compactness is actually quite difficult and technical - looking at the conditions for a function space to be compact for example. In fact just proving that the product of compact spaces is compact is non-trivial.
    It does seem like compactness is a larger topic than i thought it would be. Since i'm rapidly headed out of my league i think i'll leave it at that, thank you for the help everyone.
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