Which properties does the box topology conserve?

Question

The Wikipedia article on the Product topology has a wealth of examples of properties conserved by the product topology.

The following is a quote from the linked article:

"

Separation

• Every product of T0 spaces is T0
• Every product of T1 spaces is T1
• Every product of Hausdorff spaces is Hausdorff
• Every product of regular spaces is regular
• Every product of Tychonoff spaces is Tychonoff

Compactness Every product of compact spaces is compact (Tychonoff's theorem)

"

I know that Tychconoff's theorem doesn't hold for the box topology, i.e. the product of compact spaces with the box topology is not compact. Which, if any, of the other properties above is conserved by the product topology?

Note, that since the box topology and the product topology are identical in the finite case, all of the above are true for the box topology in the finite case.

Answer 1

It preserves the separation axioms up to Tychonoff.

In the Handbook of Set-theoretic Topology there is a chapter by Scott S. Williams on box products with the theorem:

If $X_i, i \in I$ are non-discrete, Hausdorff completely regular spaces then for an infinite index set $I$ $\prod_{i \in I} X_i$ in the box topology (also denoted $\Box_{i \in I} X_i$) is not i) locally compact or compact (ii) separable (iii) connected or locally connected (iv) first countable (v) perfect (=closed sets are $G_\delta$)

So box product are never very "nice", and a nice source of possible counterexamples. The first found ZFC Dowker space is a subspace of a box product e.g.