2020/10/15 by Ziqn Feng, Feng, Ziqn · 1 citation
Decision Sciences · Mathematics · #Advanced Topology and Set Theory #FOS: Mathematics #Fuzzy and Soft Set Theory #General Topology (math.GN) #Group Theory (math.GR) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2010.08004
openalex publication_date 2020/10/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A topological space X is defined to have a neighborhood P-base at any x∈ X from some poset P if there exists a neighborhood base (Up[x])p∈ P at x such that Up[x]⊆ Up'[x] for all p≥ p' in P. We prove that a compact space is countable, hence metrizable, if it has countable scattered height and a K(M)-base for some separable metric space M. This gives a positive answer to Problem 8.6.8 in \citeBanakh2019. 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× 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 Fréche-Urysohn topological group with a K(M)-base for some separable metric space M is first-countable, hence metrizable. And if P is a poset with calibre~(ω1, ω) 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 Cp(X) and Ck(X) are discussed. We also give an example of a topological Boolean group of character ≤ \mathfrakd such that the precompact subsets are metrizable but G doesn't have an ωω-base if ω1