On Cellular-Compact and related spaces

TL;DR

This paper investigates the properties and cardinality bounds of cellular-compact spaces, extending previous results to broader classes of spaces and answering open questions for first countable spaces.

Contribution

It provides a positive answer to whether all first countable cellular-compact spaces are weakly Lindelof and offers a new proof of a known theorem for Urysohn spaces.

Findings

First countable cellular-compact spaces are weakly Lindelof.

A new proof of Tkachuk and Wilson's theorem for Urysohn spaces.

Results extend to spaces between cellular-compact and cellular-Lindelof.

Abstract

Tkachuk and Wilson proved that a regular first countable cellular-compact space has cardinality not exceeding the continuum. In the same paper they asked if this result continues to hold for Hausdorff spaces. Xuan and Song considered the same notion and asked if every cellular-compact space is weakly Lindelof. We answer the last question for first countable spaces. As a byproduct of this result, we present a somewhat different proof of Tkachuk and Wilson theorem, valid for the wider class of Urysohn spaces. The result actually holds for a class of spaces in between cellular-compact and cellular-Lindelof. We conclude with some comments on the cardinality of a weakly linearly Lindelof space.