On $\sigma$-countably tight spaces

TL;DR

This paper investigates the cardinality of homogeneous compact spaces with certain tightness properties, extending previous results and exploring implications for product spaces, but some questions remain open.

Contribution

It extends known results on the cardinality of homogeneous compacta to $\sigma$-countably tight spaces and examines tightness conditions in product spaces.

Findings

Infinite homogeneous compacta with certain dense subspaces have cardinality continuum.

If a product space is $\sigma$-countably tight, then all but finitely many factors are countably tight.

Open question remains whether all $\sigma$-countably tight homogeneous compacta have cardinality continuum.

Abstract

Extending a result of R. de la Vega, we prove that an infinite homogeneous compactum has cardinality $\mathfrak{c}$ if either it is the union of countably many dense or finitely many arbitrary countably tight subspaces. The question if every infinite homogeneous and $\sigma$-countably tight compactum has cardinality $\mathfrak{c}$ remains open. We also show that if an arbitrary product is $\sigma$-countably tight then all but finitely many of its factors must be countably tight.