On topological groups with a small base and metrizability

Saak Gabriyelyan, Jerzy Kąkol, and Arkady Leiderman. "On topological groups with a small base and metrizability." Fundamenta Mathematicae 229.2 (2015): 129-158. <http://eudml.org/doc/283157>.

@article{SaakGabriyelyan2015,
abstract = {A (Hausdorff) topological group is said to have a -base if it admits a base of neighbourhoods of the unit, $\{U_\{α\}: α ∈ ℕ^\{ℕ\}\}$, such that $U_\{α\} ⊂ U_\{β\}$ whenever β ≤ α for all $α, β ∈ ℕ^\{ℕ\}$. The class of all metrizable topological groups is a proper subclass of the class $TG_\{\}$ of all topological groups having a -base. We prove that a topological group is metrizable iff it is Fréchet-Urysohn and has a -base. We also show that any precompact set in a topological group $G ∈ TG_\{\}$ is metrizable, and hence G is strictly angelic. We deduce from this result that an almost metrizable group is metrizable iff it has a -base. Characterizations of metrizability of topological vector spaces, in particular of $C_\{c\}(X)$, are given using -bases. We prove that if X is a submetrizable $k_\{ω\}$-space, then the free abelian topological group A(X) and the free locally convex topological space L(X) have a -base. Another class $TG_\{\}$ of topological groups with a compact resolution swallowing compact sets appears naturally. We show that $TG_\{\}$ and $TG_\{\}$ are in some sense dual to each other. We conclude with a dozen open questions and various (counter)examples.},
author = {Saak Gabriyelyan, Jerzy Kąkol, Arkady Leiderman},
journal = {Fundamenta Mathematicae},
keywords = {Frechet-Urysohn space; metrizable group; Malykhin problem; dual group; locally convex space},
language = {eng},
number = {2},
pages = {129-158},
title = {On topological groups with a small base and metrizability},
url = {http://eudml.org/doc/283157},
volume = {229},
year = {2015},
}

TY - JOUR
AU - Saak Gabriyelyan
AU - Jerzy Kąkol
AU - Arkady Leiderman
TI - On topological groups with a small base and metrizability
JO - Fundamenta Mathematicae
PY - 2015
VL - 229
IS - 2
SP - 129
EP - 158
AB - A (Hausdorff) topological group is said to have a -base if it admits a base of neighbourhoods of the unit, ${U_{α}: α ∈ ℕ^{ℕ}}$, such that $U_{α} ⊂ U_{β}$ whenever β ≤ α for all $α, β ∈ ℕ^{ℕ}$. The class of all metrizable topological groups is a proper subclass of the class $TG_{}$ of all topological groups having a -base. We prove that a topological group is metrizable iff it is Fréchet-Urysohn and has a -base. We also show that any precompact set in a topological group $G ∈ TG_{}$ is metrizable, and hence G is strictly angelic. We deduce from this result that an almost metrizable group is metrizable iff it has a -base. Characterizations of metrizability of topological vector spaces, in particular of $C_{c}(X)$, are given using -bases. We prove that if X is a submetrizable $k_{ω}$-space, then the free abelian topological group A(X) and the free locally convex topological space L(X) have a -base. Another class $TG_{}$ of topological groups with a compact resolution swallowing compact sets appears naturally. We show that $TG_{}$ and $TG_{}$ are in some sense dual to each other. We conclude with a dozen open questions and various (counter)examples.
LA - eng
KW - Frechet-Urysohn space; metrizable group; Malykhin problem; dual group; locally convex space
UR - http://eudml.org/doc/283157
ER -