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Math Help - Retract condition?

  1. #1
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    Retract condition?

    Let X be a Topological space.
    a . show that for a point a \in X there is a retract f : X \rightarrow {a}
    b . if A \subset X is finite, and X is Hausdorff with a finite amount of connected components, what is the Necessary and sufficient condition for a retract f : X \rightarrow A ?

    thanks!!
    Last edited by aharonidan; May 18th 2011 at 12:40 AM.
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  2. #2
    Senior Member Tinyboss's Avatar
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    The definition of a retract f:X\to \{a\}\subset X is a continuous function from X to {a} such that f(a)=a. And that pretty much nails it down for you.

    For (b), assuming you mean "finite" when you write "final", think about where the points of A are distributed within the components of X.
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  3. #3
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    I corrected my spelling mistake.
    I still don't get under what condition I can find a retract. Isn't the fact that both A and the connected components of X are finite is enough? I'm probably missing something...
    do all points of A need to be in the same component of X?
    thanks
    Last edited by aharonidan; May 18th 2011 at 01:00 AM.
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  4. #4
    Senior Member Tinyboss's Avatar
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    Think about how f would act on a path connecting distinct points of A.
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