A nice subclass of functionally countable spaces

Abstract

A space $X$ is {functionally countable} if $f(X)$ is countable for any continuous function $f\colon X \to {\mathbb{R}}$. We will call a space $X$ {exponentially separable} if for any countable family ${\mathcal{F}}$ of closed subsets of $X$, there exists a countable set $A\subset X$ such that $A\cap \bigcap {\mathcal{G}}\neq\emptyset$ whenever ${\mathcal{G}}\subset {\mathcal{F}}$ and $\bigcap {\mathcal{G}}\neq\emptyset$. Every exponentially separable space is functionally countable; we will show that for some nice classes of spaces exponential separability coincides with functional countability. We will also establish that the class of exponentially separable spaces has nice categorical properties: it is preserved by closed subspaces, countable unions and continuous images. Besides, it contains all Lindelof $P$-spaces as well as some wide classes of scattered spaces. In particular, if a scattered space is either Lindelof or ${\omega}$-bounded, then it is exponentially separable.