Spaces C(X)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⊆NNΣ⊆NN is unbounded and directed (and if additionally each subset of Σ which is bounded in NNNN has a bound at Σ) a base {Uα:α∈Σ}{Uα:α∈Σ} of neighborhoods of the identity of a topological group G   with Uβ⊆UαUβ⊆Uα whenever α≤βα≤β with α,β∈Σα,β∈Σ is called a Σ-base (a Σ2Σ2-base). The case when Σ=NNΣ=NN has been noticed for topological vector spaces (under the name of GG-base) at [2]. If X   is a separable and metrizable space which is not Polish, the space Cc(X)Cc(X) has a Σ-base but does not admit any GG-base. A topological group which is Fréchet–Urysohn is metrizable iff it has a Σ2Σ2-base of the identity. Under an appropriate ZFC model the space Cc(ω1)Cc(ω1) has a Σ2Σ2-base which is not a GG-base. We also prove that (i  ) every compact set in a topological group with a Σ2Σ2-base of neighborhoods of the identity is metrizable, (ii)(ii) a Cp(X)Cp(X) space has a Σ2Σ2-base iff X   is countable, and (iii)(iii) if a space Cc(X)Cc(X) has a Σ2Σ2-base then X is a C  -Suslin space, hence Cc(X)Cc(X) is angelic.