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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
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
Summary:
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, Lindelöf 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.
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