Spaces C (X) with ordered bases

The concept of Sigma-base of neighborhoods of the identity of a topological group G is introduced. If the index set Sigma subset of N-N is unbounded and directed (and if additionally each subset of Sigma which is bounded in N-N has a bound at Sigma) a base {U-alpha : alpha is an element of Sigma} of neighborhoods of the identity of a topological group G with U-beta subset of U-alpha whenever alpha <= beta with alpha, beta is an element of Sigma is called a Sigma-base (a Sigma(2)-base). The case when Sigma = 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 Sigma-base but does not admit any G-base. A topological group which is Frechet-Urysohn is metrizable iff it has a Sigma(2)-base of the identity. Under an appropriate ZFC model the space C-c (omega(1)) has a Sigma(2)-base which is not a G-base. We also prove that (i) every compact set in a topological group with a Sigma(2)-base of neighborhoods of the identity is metrizable, (ii) a C-p (X) space has a Sigma(2)-base iff X is countable, and (iii) if a space C-c(X) has a Sigma(2)-base then X is a C-Suslin space, hence C-c(X)is angelic. (C) 2016 Elsevier B.V. All rights reserved.