$P$-bases and topological groups
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Abstract:
A topological space $X$ is defined to have a neighborhood $P$-base at any $x\in X$ from some partially ordered set (poset) $P$ if there exists a neighborhood base $(U_p[x])_{p\in P}$ at $x$ such that $U_p[x]\subseteq U_{p’}[x]$ for all $p\geq p’$ in $P$. We prove that a compact space is countable, hence metrizable, if it has countable scattered height and a $\mathcal {K}(M)$-base for some separable metric space $M$. Banakh [Dissertationes Math. 538 (2019), p. 141] gives a positive answer to Problem 8.6.8.
Let $A(X)$ be the free Abelian topological group on $X$. It is shown that if $Y$ is a retract of $X$ such that the free Abelian topological group $A(Y)$ has a $P$-base and $A(X/Y)$ has a $Q$-base, then $A(X)$ has a $P\times Q$-base. Also if $Y$ is a closed subspace of $X$ and $A(X)$ has a $P$-base, then $A(X/Y)$ has a $P$-base.
It is shown that any Fréchet-Urysohn topological group with a $\mathcal {K}(M)$-base for some separable metric space $M$ is first-countable, hence metrizable. And if $P$ is a poset with calibre $(\omega _1, \omega )$ and $G$ is a topological group with a $P$-base, then any precompact subset in G is metrizable, hence $G$ is strictly angelic. Applications in function spaces $C_p(X)$ and $C_k(X)$ are discussed. We also give an example of a topological Boolean group of character $\leq \mathfrak {d}$ such that the precompact subsets are metrizable but $G$ doesn’t have an $\omega ^\omega$-base if $\omega _1<\mathfrak {d}$. Gabriyelyan, Kakol, and Liederman [Fund. Math. 229 (2015), pp. 129–158] give a consistent negative answer to Problem 6.5.
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Additional Information
- Ziqin Feng
- Affiliation: Department of Mathematics and Statistics, Auburn University, Auburn, Alabama 36849
- MR Author ID: 803357
- Email: zzf0006@auburn.edu
- Received by editor(s): October 21, 2020
- Received by editor(s) in revised form: February 26, 2021, and May 19, 2021
- Published electronically: December 1, 2021
- Communicated by: Heike Mildenberger
- © Copyright 2021 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 877-889
- MSC (2020): Primary 22A05, 54H11, 46A50
- DOI: https://doi.org/10.1090/proc/15671
- MathSciNet review: 4356194