1. ## 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 don´t want the exact answers, but some hints, for I am totally clueless but I want to think for myself...
Thanks very much!

2. Originally Posted by Inti
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.

Originally Posted by Inti
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.

Originally Posted by Inti
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.)