On the reflection of the countable chain condition

TL;DR

This paper investigates conditions under which uncountable ccc topological spaces contain smaller ccc subspaces, providing new examples and extending understanding of the reflection properties of the countable chain condition.

Contribution

It establishes that compact Hausdorff and certain Hausdorff spaces with low pseudocharacter contain ccc subspaces of size , and constructs new examples of ccc spaces lacking smaller ccc subspaces.

Findings

Uncountable ccc compact Hausdorff spaces contain -sized ccc subspaces.

Constructed ccc Tychonoff spaces of size  with no smaller ccc subspaces.

Provided a ccc compact T1 space of size  without smaller ccc subspaces.

Abstract

We study the question of when an uncountable ccc topological space $X$ contains a ccc subspace of size $\aleph_1$. We show that it does if $X$ is compact Hausdorff and more generally if $X$ is Hausdorff with $\mathrm{pct}(X) \leq \aleph_1$. For each regular cardinal $\kappa$, an example is constructed of a ccc Tychonoff space of size $\kappa$ and countable pseudocharacter but with no ccc subspace of size less than $\kappa$. We also give a ccc compact $T_1$ space of size $\kappa$ with no ccc subspace of size less than $\kappa$.