Let $(X,\tau)$ be a topological space. Let $\text{Cont}(X,X)$ denote the set of continuous functions $f:X\to X$. 


What can be said about spaces $(X,\tau)$ where $|\text{Cont}(X,X)| = |X|$? For instance, is it impossible that they are [zero-dimensional](http://en.wikipedia.org/wiki/Zero-dimensional_space)? (Note that trivially spaces with $|\text{Cont}(X,X)| = |X|$ must be infinite, because all constant functions as well as the identity are members of $\text{Cont}(X,X)$.)



Topological spaces $(X,\tau)$ where $|\text{Cont}(X,X)| = |X|$
If a Hausdorff space $\ X\ $ admits a dense subset $ A\subseteq X\ $ such that


$$|X|^{|A|}\ =\ |X|$$


then indeed $\ \left|Cont(X\ X)\right|\ =\ |X|$.


This holds in particular for the separable metric spaces of cardinality continuum, as was already noted by Tomek Kania in the first ***Answer***.


