Remarks on countable tightness

TL;DR

This paper characterizes when countable tightness in topological spaces is preserved under countably closed forcing, using combinatorial conditions, and explores its behavior in generic extensions and for stronger variants.

Contribution

It provides a combinatorial characterization of indestructibility of countable tightness under countably closed forcing and analyzes its behavior in Cohen real extensions.

Findings

Certain classes of spaces are indestructibly countably tight

Countable tightness can be destroyed by countably closed forcing

Stronger variants like selective separability are also examined

Abstract

Countable tightness may be destroyed by countably closed forcing. We characterize the indestructibility of countable tightness under countably closed forcing by combinatorial statements similar to the ones Tall used to characterize indestructibility of the Lindelof property under countably closed forcing. We consider the behavior of countable tightness in generic extensions obtained by adding Cohen reals. We show that certain classes of well-studied topological spaces are indestructibly countably tight. Stronger versions of countable tightness, including selective versions of separability, are further explored.