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

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

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!