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Thread: retract in a topological space

  1. #1
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    retract in a topological space

    If X is a topological space, what would be an example of a retract of X which is not closed?

    I know that if X is Hausdorff, every retract of X must be closed. So I think we need to come up with a topological space X which is not Hausdorff.

    [A C X is a retract of X if there exists a continuous map r:X-->A such that r(y)=y for each y in A].
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  2. #2
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    Quote Originally Posted by math8 View Post
    If X is a topological space, what would be an example of a retract of X which is not closed?

    I know that if X is Hausdorff, every retract of X must be closed. So I think we need to come up with a topological space X which is not Hausdorff.

    [A C X is a retract of X if there exists a continuous map r:X-->A such that r(y)=y for each y in A].
    Let T be a finite complement topology on $\displaystyle Z_+$. We know that $\displaystyle 2Z_+$ is not closed in $\displaystyle (Z_+, T)$ (2$\displaystyle Z_+$ is not finite in $\displaystyle Z_+$).

    Let r be a function $\displaystyle r:Z_+ \rightarrow 2Z_+$ such that r(2i-1)=r(2i)=2i for each $\displaystyle i \in Z_+$. r is well-defined. For any closed subset A of $\displaystyle 2Z_+$, $\displaystyle r^{-1}(A)$ is closed in $\displaystyle (Z_+, T)$. Thus r is continuous. We also have r(2i)=2i for each i in $\displaystyle Z_+$.

    Thus, 2$\displaystyle Z_+$ is a retract of $\displaystyle Z_+$, which is not closed in $\displaystyle (Z_+, T)$.
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