On first and second countable spaces and the axiom of choice (Q1880711)

This paper discusses the role of the axiom of countable choice for families of sets of real numbers in first and second countable spaces. All spaces are considered in ZF. Denote by \(\text{CC}(\mathbb{R})\) the axiom of countable choice for families of sets of real numbers. The following concept is introduced: A topological space is called super second countable (SSC) if each base has a countable subfamily which is a base. The author proves that the following propositions are equivalent: (1) \(\text{CC}(\mathbb{R})\), (2) the axiom of countable choice is true for families of dense subspaces of \(\mathbb{R}\), (3) every subspace of \(\mathbb{R}\) is separable, (4) every dense subspace of \(\mathbb{R}\) is separable, (5) every base of a second countable space has a countable subfamily which is a base, (6) every base of open sets of \(\mathbb{R}\) has a countable subfamily which is a base, (7) \(\mathbb{R}\) is SSC, (8) every separable metric space is SSC. Some questions are mentioned: (1) Is SSC hereditary? (2) Are there non-separable SSC-spaces? (3) Are there uncountable SSC \(T_0\)-spaces? For the first countable case, two conditions are stated: one of them classical, namely CUT: the countable union of countable sets is countable. Another is a recent statement: \(\omega\)-MC: for each family \((X_i)_{i\in I}\) of non-empty sets, there is a family \((A_i)_{i\in I}\) such that each \(A_i\) is non-empty, finite or countable infinite and \(A_i\subset X_i\) for all \(i\in I\). Then the following are equivalent: (1) \(\omega\)-MC and CC, (2) \(\omega\)-MC and CUT, (3) In a first countable space, for each local base \(({\mathfrak B}(x))_{x\in X}\), there is a local base \(({\mathfrak V}(x))_{x\in X}\), where for each \(x\in X\), \({\mathfrak V}(x)\) is countable, and \({\mathfrak V}(x)\subset{\mathfrak B}(x)\). Moreover, some results are mentioned. If the countable product of second countable spaces is second countable, CUT holds. CC holds for families of sets with at most \(2^{\aleph_0}\), the countable product of second countable spaces is second countable, \(\text{CC}(\text{CC}\mathbb{R}))\Leftrightarrow\) a first (second) countable space is Hausdorff if and only if each sequence has at \(n\)-most one limit.