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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 Tikhonov space of size $\kappa $ and countable pseudocharacter but with no uncountable ccc subspace of size less than $\kappa $. We also give a ccc compact $T_1$ space of size $\kappa $ with no uncountable ccc subspace of size less than $\kappa $.