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Homeomorphism example

  1. Apr 21, 2010 #1
    Hi Guy's
    I need to show that two spaces are Homeomorphic for a given function between them.
    Is there an online example of a proof.

    A lot of text on the web tells you what it needs to be a homeomorphism but I not an example of a proof. I just want an good example I can you to help me.

    Thanks in advance
     
  2. jcsd
  3. Apr 22, 2010 #2
    The definition of homeomorphism (map is continuous, as is its inverse) is also the strategy for the proof.
     
  4. Apr 22, 2010 #3
    What you gotten down so far? Is the domain connected? The codomain Hausdorff?
     
    Last edited: Apr 22, 2010
  5. Apr 23, 2010 #4
    What I've got so far is...

    I must say in advance that this is an assignment question.

    I have been given the following.

    Let [itex](X,T)[/itex] be a topological space. Let I := [itex][0,1]:= {t \in \Real \mid 0 \leq t \leq 1} [/itex]
    be endowed with the Euclidean topology. Prove that for each [itex]\lambda \in [0,1][/itex] the function:

    [itex] i_{\lambda}: X \rightarrow X \times I, x \rightarrow(x,\lambda)[/itex]

    is a homeomorphism of X onto [itex]im(i_{\lambda})[/itex], where [itex]X \times I[/itex] is endowed with the product topology.

    I know that if two spaces are homeomorhic you need a fucntion between the spaces that satisfy.


    1: F is one-one
    2: F is onto
    3; A subset [itex]A \subset X[/itex]is open if and only if [itex] f(A)[/itex] is open.



    Therfore we need to show that the inverse function [itex]i_{\lambda}^{-1}(t_0 \times \sigma_{\lambda})[/itex] is open in A whenever

    [itex](t_0 \times \sigma_{\lambda})[/itex] is open in [itex]X \times I[/itex] where [itex]t \in T[/itex]


    but [itex](t_0 \times \sigma_{\lambda})[/itex] open implies [itex]t_0 \in T[/itex]
    , [itex]\sigma_{\lambda} = [\lambda - \epsilon_{1},\lambda - \epsilon_{2}}[/itex] and [itex]i_{\lambda}^{-1}(t_0 \times \sigma_{\lambda}) = t_0[/itex]

    Since [itex](x,\lambda) \in t_0 \times \sigma_{\lambda i}) [/itex]implies [itex]x \in t_0 , \lambda \in \sigma_{\lambda} [/itex]

    Therefore [itex]i_{\lambda}[/itex] is a homeomorphism
     
  6. Apr 24, 2010 #5

    I think it may be clearer if you invert the order here: in order to show that
    (X,TX) and (Y,TY are homeomorphic to each other,
    you must find a function f so that :

    1) f is continuous

    2)f^-1 is also continuous.

    From these it follows that f has to be bijective. So in this case, first show continuity
    of f :

    1)take an open set in XxI product ( or take a basic or subbasic open set, easier)

    and show its inverse image is open in X . Then show that f-1 is also

    continuous; like you said, this implies that if you take any O open in X , then

    f-1(O) must be open in the product space XxI


    HTH.
     
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