If f maps X to X continuously and the range of f is A (subset of X) then is this sufficient to show that there exist a continuous retraction from X to A?
If so, how can this be proven?
If not, any counterexamples?
Notes: This is the topological continuity I am speaking of. Also a I am using retraction to mean a map from X to a subspace A such that f(a)=a for all a in A.
What I mean is that the the existence of continuous map where should guarantee the existence of a continuous retraction of to .
Actually I'd like to make my statement even stronger:
If there is a function where then there exists a continuous retraction such that .
Is this true? Proof?
Is it false? Counterexample?