Spaces C(X) with ordered bases

The concept of Σ-base of neighborhoods of the identity of a topological group G is introduced. If the index set Σ ⊆ N N is unbounded and directed (and if additionally each subset of Σ which is bounded in N N has a bound at Σ) a base { U α : α ∈ Σ } of neighborhoods of the identity of a topological group G with U β ⊆ U α whenever α ≤ β with α , β ∈ Σ is called a Σ-base (a Σ 2 -base). The case when Σ = N N 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 C c ( X ) has a Σ-base but does not admit any G -base. A topological group which is Frechet–Urysohn is metrizable iff it has a Σ 2 -base of the identity. Under an appropriate ZFC model the space C c ( ω 1 ) has a Σ 2 -base which is not a G -base. We also prove that (i) every compact set in a topological group with a Σ 2 -base of neighborhoods of the identity is metrizable, ( i i ) a C p ( X ) space has a Σ 2 -base iff X is countable, and ( i i i ) if a space C c ( X ) has a Σ 2 -base then X is a C-Suslin space, hence C c ( X ) is angelic.