Given a topological space $X$, let $X_\delta$ be the topology on $X$ generated by the $G_\delta$ subsets of $X$. Let $c(X)$ be the cellularity of $X$, that is, the supremum of cardinalities of families of pairwise disjoint non-empty open subsets $X$. Thus $X$ is ccc if and only if $c(X)=\aleph_0$.

In 1972 Juhász proved:

Let $X$ be a compact Hausdorff space. Then $c(X_\delta) \leq 2^{c(X)}$.

Compactness can actually be replaced with pseudocompactness+regularity in Juhász's result.

Some form of compactness is essential in Juhász's theorem. For every cardinal $\kappa$ there is a (regular) ccc space $X$ such that $c(X_\delta) > \kappa$; some examples of such a space can be found in this other Mathoverflow question of mine. However, none of those examples is Lindelof, as they all contain large closed discrete subsets. This motivates the following question:

QUESTION: Let $\kappa$ be any cardinal. Is there a Lindelof regular ccc space such that $c(X_\delta) > \kappa$?

To find a consistent example of a Lindelof ccc regular space such that $c(X_\delta) > 2^{\aleph_0}$ it suffices to take a small modification of Gorelic's example of a Lindelof space with points $G_\delta$ and cardinality larger than the continuum (see the remark at the beginning of page 606 of Gorelic's paper).

A related Mathoverflow question is: Covering compact Hausdorff spaces with closed $G_\delta$ sets



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.