When does the 0-dimensional Reflection Preserve Products?

Andrej Bauer

A Question about Zero-dimensional Spaces

We say that a topological space is zero-dimensional when the collection of its clopen (closed and open) subsets forms a basis for its topology.

For a topological space X, let z(X) be its zero-dimensional reflection, i.e., the same underlying set X but with the topology generated by the collection of the clopen subsets of X.

Question 1: Does z preserve products, i.e., is z(X ×Y) homeomorphic to z(X) ×z(Y)?

The question can be stated equivalently as follows.

Question 1': Is every clopen subset of X ×Y a union of the sets of the form U ×V, where U 'subset or equal' X is clopen and V 'subset or equal' Y is clopen?