Point-countable π-bases in first countable and similar spaces

V. V. Tkachuk. "Point-countable π-bases in first countable and similar spaces." Fundamenta Mathematicae 186.1 (2005): 55-69. <http://eudml.org/doc/283150>.

@article{V2005,
abstract = {It is a classical result of Shapirovsky that any compact space of countable tightness has a point-countable π-base. We look at general spaces with point-countable π-bases and prove, in particular, that, under the Continuum Hypothesis, any Lindelöf first countable space has a point-countable π-base. We also analyze when the function space $C_\{p\}(X)$ has a point-countable π -base, giving a criterion for this in terms of the topology of X when l*(X) = ω. Dealing with point-countable π-bases makes it possible to show that, in some models of ZFC, there exists a space X such that $C_\{p\}(X)$ is a W-space in the sense of Gruenhage while there exists no embedding of $C_\{p\}(X)$ in a Σ-product of first countable spaces. This gives a consistent answer to a twenty-years-old problem of Gruenhage.},
author = {V. V. Tkachuk},
journal = {Fundamenta Mathematicae},
keywords = {first countable space; -space; point-countable -base; winning strategy; scattered space; Lindelöf -space},
language = {eng},
number = {1},
pages = {55-69},
title = {Point-countable π-bases in first countable and similar spaces},
url = {http://eudml.org/doc/283150},
volume = {186},
year = {2005},
}

TY - JOUR
AU - V. V. Tkachuk
TI - Point-countable π-bases in first countable and similar spaces
JO - Fundamenta Mathematicae
PY - 2005
VL - 186
IS - 1
SP - 55
EP - 69
AB - It is a classical result of Shapirovsky that any compact space of countable tightness has a point-countable π-base. We look at general spaces with point-countable π-bases and prove, in particular, that, under the Continuum Hypothesis, any Lindelöf first countable space has a point-countable π-base. We also analyze when the function space $C_{p}(X)$ has a point-countable π -base, giving a criterion for this in terms of the topology of X when l*(X) = ω. Dealing with point-countable π-bases makes it possible to show that, in some models of ZFC, there exists a space X such that $C_{p}(X)$ is a W-space in the sense of Gruenhage while there exists no embedding of $C_{p}(X)$ in a Σ-product of first countable spaces. This gives a consistent answer to a twenty-years-old problem of Gruenhage.
LA - eng
KW - first countable space; -space; point-countable -base; winning strategy; scattered space; Lindelöf -space
UR - http://eudml.org/doc/283150
ER -