$\text{Cont}(X,X)$ and $\neg\mathsf{GCH}$

Question

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 $|\text{Cont}(X,X)| = |X|$.)

Answer 1

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 $2^{\aleph_0}=\aleph_{\omega_1}$.