Some applications of exponentially separable spaces

Abstract

We prove that any finite product of functionally countable spaces is functionally countable if it has the Lindelöf property. In particular, any finite product of functionally countable Lindelöf Σ-spaces is functionally countable. A Lindelöf Σ- space is functionally countable if and only if it is exponentially separable. We give an example of an exponentially separable space whose square is not exponentially separable and establish that a finite-like space in the sense of Telgarsky must be exponentially separable. Finally, we introduce the class of weakly exponentially separable spaces and show that all countably compact spaces and all Lindelöf Σ-spaces belong to this class. Our results solve several published open questions.