On first and second countable spaces and the axiom of choice

Abstract

In this paper it is studied the role of the axiom of choice in some theorems in which the concepts of first and second countability are used. Results such as the following are established: (1) In ZF (Zermelo–Fraenkel set theory without the axiom of choice), equivalent are: (i) every base of a second countable space has a countable subfamily which is a base; (ii) the axiom of countable choice for sets of real numbers. (2) In ZF, equivalent are: (i) every local base at a point x, in a first countable space, contains a countable base at x; (ii) the axiom of countable choice ( CC). (3) In ZF, equivalent are: (i) for every local base system ( B(x)) x∈X of a first countable space X, there is a local base system ( V(x)) x∈X such that, for each x∈ X, V(x) is countable and V(x)⊆ B(x) ; (ii) for every family ( X i ) i∈ I of non-empty sets there is a family ( A i ) i∈ I of non-empty, at most countable sets, such that A i ⊆ X i for every i∈ I ( ω- MC) and CC.