On first and second countable spaces and the axiom of choice

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: (Zermelo–Fraenkel set theory without the axiom of choice), equivalent are: (i) base of a second countable space has a countable subfamily which is a base; iom of countable choice for sets of real numbers. equivalent are: (i) local base at a point x, in a first countable space, contains a countable base at x; iom of countable choice (CC). equivalent are: (i) ery 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); ery family (Xi)i∈I of non-empty sets there is a family (Ai)i∈I of non-empty, at most countable sets, such that Ai⊆Xi for every i∈I (ω-MC) and CC.