is called a retract if there exists a continuous function (the retraction) such that for every

i) Prove that is a retract of iff for every topological space , every continuous function extends to a continuous function in .

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!