For a topological space $(X,\tau)$ let $\text{Cont}(X,X)$ denote the set of continuous functions $f:X\to X$. Is it consistent that there a space $(X,\tau)$ such that $$|X| < |\text{Cont}(X,X)| < 2^{|X|}?$$


(Also see [Topological spaces $(X,\tau)$ where $|\text{Cont}(X,X)| = |X|$](https://mathoverflow.net/questions/196371/topological-spaces-x-tau-where-textcontx-x-x).)



$\text{Cont}(X,X)$ and $\neg\mathsf{GCH}$
Choose a set $S\subseteq\mathbb R$ with $|S|=\aleph\_1$ and let $X=\mathbb N\times S$. Then $|X|=\aleph\_1,\ 2^{|X|}=2^{\aleph\_1}$, and $|\text{Cont}(X,X)|=2^{\aleph\_0}$. It is consistent that $\aleph\_1\lt2^{\aleph\_0}\lt2^{\aleph\_1}$; e.g., just make $2^{\aleph\_0}=\aleph\_{\omega\_1}$.


