Abstract:A space $X$ is functionally countable (FC) if for every continuous $f:X\to \mathbb R$, $|f(X)|\leq \omega$. The class of FC spaces includes ordinals, some trees, compact scattered spaces, Lindel\"of P-spaces, $\sigma$-products in $2^\kappa$, and some L-spaces. We consider the following three versions of functional separability: $X$ is 1-FS if it has a dense FC subspace; $X$ is 2-FS if there is a dense subspace $Y\subset X$ such that for every continuous $f:X\to \mathbb R$, $|f(Y)|\leq\omega$; $X$ is 3-FS if for every continuous $f:X\to \mathbb R$, there is a dense subspace $Y\subset X$ such that $|f(Y)|\leq \omega$. We give examples distinguishing 1-FS, 2-FS, and 3-FS and discuss some properties of functionally separable spaces.
Keywords: functionally countable, pseudo-$\aleph_1$-compact, DCCC, P-space, $\tau$-simple, scattered, 1-functionally separable, 2-functionally separable, 3-functionally separable, pseudocompact, dyadic compactum, $\sigma$-centered base, LOTS
AMS Subject Classification: 54C30 54D65