Functional countability and exponential separability of product spaces and subspaces

TL;DR

This paper explores properties like functional countability and exponential separability in product spaces, solving open problems and extending known results in topology.

Contribution

It proves that the product of functionally countable pseudocompact spaces remains functionally countable and extends exponential separability results to $\sigma$-products of non-zero ordinals.

Findings

Product of functionally countable pseudocompact spaces is functionally countable

Hereditary Lindel"ofness of certain spaces is independent of ZFC

$\sigma$-product of non-zero ordinals is exponentially separable

Abstract

We investigate the behavior of functional countability and exponential separability in products and subspaces of topological spaces. We solve a problem of Tkachuk by showing that the product of functionally countable pseudocompact spaces is itself functionally countable. Solving another problem of Tkachuk, we show that it is independent of ZFC whether regular spaces which have all their subspaces functionally countable are hereditarily Lindel\"of. Finally, we prove that the $\sigma$-product of non-zero ordinals is exponentially separable, thereby extending a result of Kemoto and Szeptycki.