$P$-bases and Topological Groups

Abstract

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] ⊆ U p ′ [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 [3].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 Fréchet-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 (ω 1 , 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 C k (X) are discussed. We also give an example of a topological Boolean group of character ≤ d such that the precompact subsets are metrizable but G doesn't have an ω ω -base if ω 1 < d.