This is probably a very basic question, but I don't know the answer and I also don't see how it might be obvious (which it very well might be).

Given a topological space $X$, when is the set of all its continuous self-maps generated by the subset consisting of

• the self-hemeomorphisms of prime order (i.e., there exists some prime number $p$ such that $f^p = id$ while $f^k \neq id$ for all $k \leq p$) and
• the continuous retractions (i.e. $f^2 = f$)?

Is the answer to this a well-known fact? Is there a chracterization of the cases where it is true? All I know is that it is true if the topology is discrete or trivial (so we are essentially talking about sets and functions), although is think that it requires AC.

I should add that, by generation, I mean everything you can get by applying composition finitely many times.

Edit: I originally also asked whether the statement is true or not. A counterexample was quickly presented by James Cranch (see below), so I have put more emphasis on a different part of my question, namely if there is a characterization of the cases where the statement is true.