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Math Help - Retraction

  1. #1
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    Retraction

    A \subset X is called a retract if there exists a continuous function r : X \longrightarrow A (the retraction) such that r(a)=a for every a \in A


    i) Prove that A is a retract of X iff for every topological space Y, every continuous function f : A \longrightarrow Y extends to a continuous function in X.

    ii) Prove that a retract of a hausdorff space is a closed set.

    iii) Let X be an infinite set with the finite complements topology. Prove that every nonempty open set is a retract of X.


    I donīt want the exact answers, but some hints, for I am totally clueless but I want to think for myself...
    Thanks very much!
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  2. #2
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    Quote Originally Posted by Inti View Post
    i) Prove that A is a retract of X iff for every topological space Y, every continuous function f : A \longrightarrow Y extends to a continuous function in X.
    For the forward implication, consider the composition r\circ f. For the reverse implication, take Y=A and f to be the identity map.

    Quote Originally Posted by Inti View Post
    ii) Prove that a retract of a hausdorff space is a closed set.
    Show that for a continuous map r, the set \{x\in X:r(x)=x\} is closed.

    Quote Originally Posted by Inti View Post
    iii) Let X be an infinite set with the finite complements topology. Prove that every nonempty open set is a retract of X.
    A nonempty open subset A of X has a finite complement. Define the retraction map to be the identity on A, and let it map the finitely many elements not in A to some random element of A. (You then have to prove that this map is continuous. In other words, if a set has a finite complement, then its inverse image under this map should also have a finite complement.)
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