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Math Help - Topology Question...

  1. #1
    TTB
    TTB is offline
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    Topology Question...

    Hi guys,

    I really need help with the following question:

    Let A be the infinite countable union of circles with radius 1/n & centres (1/n,0) in Euclidean 2-space. ie - subspace

    Let B be the infinite countable union of circles with radius n & centres (n,0) in Euclidean 2-space. e - subspace

    Let C be the wedge of countably many circles.

    1) Which of these 3 spaces are CW complexes?

    2) Attempt to construct contiinuous bijectiive maps between these spaces (if possible) & then show if these are homeomorphisms.

    3) Are the any other more "immediate" way of showing any of the spaces are homeomorphic?

    I'm completely stuck! :-( Any assistance would be ENORMOUSLY helpful!

    Thanks in advance. :-) x
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  2. #2
    TTB
    TTB is offline
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    PS - the instructor hinted that A is not homeomorphic to B or C - but I have no idea how to show this! :-s :-(

    Thanks x
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  3. #3
    Senior Member
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    Quote Originally Posted by TTB View Post
    Hi guys,

    I really need help with the following question:

    Let A be the infinite countable union of circles with radius 1/n & centres (1/n,0) in Euclidean 2-space. ie - subspace

    Let B be the infinite countable union of circles with radius n & centres (n,0) in Euclidean 2-space. e - subspace

    Let C be the wedge of countably many circles.

    1) Which of these 3 spaces are CW complexes?

    2) Attempt to construct contiinuous bijectiive maps between these spaces (if possible) & then show if these are homeomorphisms.

    3) Are the any other more "immediate" way of showing any of the spaces are homeomorphic?

    I'm completely stuck! :-( Any assistance would be ENORMOUSLY helpful!

    Thanks in advance. :-) x
    Every CW complex is locally contractible, but your space A is not locally contractible at 0. A is also compact. Remember that "compactness" is a topological invariant. If one space is compact and the other space is not compact, we can't find a homeomorphism between them.
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