Functional separability

Levy, Ronnie, and Matveev, M.. "Functional separability." Commentationes Mathematicae Universitatis Carolinae 51.4 (2010): 705-711. <http://eudml.org/doc/246508>.

@article{Levy2010,
abstract = {A space $X$ is functionally countable (FC) if for every continuous $f:X\rightarrow \mathbb \{R\}$, $|f(X)|\le \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 $f:X\rightarrow \mathbb \{R\}$, $|f(Y)|\le \omega $; $X$ is 3-FS if for every continuous $f:X\rightarrow \mathbb \{R\}$, there is a dense subspace $Y\subset X$ such that $|f(Y)|\le \omega $. We give examples distinguishing 1-FS, 2-FS, and 3-FS and discuss some properties of functionally separable spaces.},
author = {Levy, Ronnie, Matveev, M.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {functionally countable; pseudo-$\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 space; functionally separable space; pseudocompact space},
language = {eng},
number = {4},
pages = {705-711},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Functional separability},
url = {http://eudml.org/doc/246508},
volume = {51},
year = {2010},
}

TY - JOUR
AU - Levy, Ronnie
AU - Matveev, M.
TI - Functional separability
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2010
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 51
IS - 4
SP - 705
EP - 711
AB - A space $X$ is functionally countable (FC) if for every continuous $f:X\rightarrow \mathbb {R}$, $|f(X)|\le \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 $f:X\rightarrow \mathbb {R}$, $|f(Y)|\le \omega $; $X$ is 3-FS if for every continuous $f:X\rightarrow \mathbb {R}$, there is a dense subspace $Y\subset X$ such that $|f(Y)|\le \omega $. We give examples distinguishing 1-FS, 2-FS, and 3-FS and discuss some properties of functionally separable spaces.
LA - eng
KW - functionally countable; pseudo-$\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 space; functionally separable space; pseudocompact space
UR - http://eudml.org/doc/246508
ER -