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Assuming countable choice, a countable product of second countable spaces is second countable
Statement
Assuming , a countable product of second countable spaces is second countable.
Facts & Assumptions
Given: Second countable factors indexed by a countable set.
Countable choice selects a countable basis in each factor (The Axiom of Countable Choice ()).
A product of two at most countable sets is at most countable, and under countable choice a countable union of at most countable sets is at most countable (A product of two at most countable sets is at most countable, Countable unions of at most countable sets, assuming ).
Finite-support boxes whose nontrivial coordinates are basis members form a basis for the product topology (The product set of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space, Second countability: an at most countable basis for the topology).
Proof
Use [A1] to choose countable factor bases.
Finite-support boxes with selected basic coordinates form a basis for the product by [F1].
The finite subsets of a countable index set form an at most countable family: after enumerating the index set, subsets of size at most are coded by -tuples of natural numbers, which are countable by finite induction using the product theorem in [L1], and their union over is countable by the union theorem in [L1]. For each fixed finite support , the choices of one member of the selected basis in every coordinate of form a finite product of countable sets and are countable by the same induction. A final application of the countable-union theorem shows that all finite-support boxes form an at most countable family.
Thus the product is second countable.
Depends on
Second countability: an at most countable basis for the topology
The product set of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space
A product of two at most countable sets is at most countable
Countable unions of at most countable sets, assuming
The Axiom of Countable Choice ()