TITLE: A rare property of Hausdorff spaces
QUESTION [25 upvotes]: Is there a Hausdorff topological space $X$ such  that for any continuous map $f: X\longrightarrow \mathbb{R}$ and any $x\in \mathbb{R}$, the set $f^{-1}(x)$ is either empty or infinite?

REPLY [3 votes]: If $(X,\mathcal{S})$ is a regular space, then the $P$-space coreflection of $(X,\mathcal{S})$ is the space generated by the basis $\{\bigcap_{n\in\omega}U_{n}\mid\forall n\,U_{n}\in\mathcal{S}\}$. A completely regular space $X$ satisfies the property that $f^{-1}(\{x\})$ is empty or infinite for any real $x$ if and only if the $P$-space coreflection of $X$ has no isolated points. 
There exists infinite regular spaces $X$ where every continuous function $f:X\rightarrow\mathbb{R}$ is constant and in such spaces $f^{-1}(x)$ is always empty or infinite.