Spaces [formula omitted] with ordered bases

Abstract

The concept of Σ-base of neighborhoods of the identity of a topological group G is introduced. If the index set Σ⊆NN is unbounded and directed (and if additionally each subset of Σ which is bounded in NN has a bound at Σ) a base {Uα:α∈Σ} of neighborhoods of the identity of a topological group G with Uβ⊆Uα whenever α≤β with α,β∈Σ is called a Σ-base (a Σ2-base). The case when Σ=NN has been noticed for topological vector spaces (under the name of G-base) at [2]. If X is a separable and metrizable space which is not Polish, the space Cc(X) has a Σ-base but does not admit any G-base. A topological group which is Fréchet–Urysohn is metrizable iff it has a Σ2-base of the identity. Under an appropriate ZFC model the space Cc(ω1) has a Σ2-base which is not a G-base. We also prove that (i) every compact set in a topological group with a Σ2-base of neighborhoods of the identity is metrizable, (ii) a Cp(X) space has a Σ2-base iff X is countable, and (iii) if a space Cc(X) has a Σ2-base then X is a C-Suslin space, hence Cc(X) is angelic.