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A space $X$ is called an almost $P$-space if $Int(G) \neq \emptyset$ for every non-empty $G_\delta$ subset $G \subset X$.

Every $P$-space (that is, a space where $G_\delta$s are open) is an almost $P$-space, but the class of almost $P$-spaces is much wider and even contains some compact spaces like $\omega^*$.

In his answer to another question, Joseph van Name notes that no infinite compact group can be an almost $P$-space (this is an easy consequence of the existence of a Haar probability measure). Joseph's observation suggests a natural question:

QUESTION: Is there an infinite compact Hausdorff homogeneous almost $P$-space?

Ronnie Levy notes that every compact linearly ordered almost $P$-space has a $P$-point, hence no example answering the above question can be a linearly ordered space (if $X$ is homogeneous and has a $P$-point then every point of $X$ must be a $P$-point, but every infinite compact space must have a non-$P$-point).

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  • $\begingroup$ An almost P-space has uncountable cellularity and because of that cannot be homeomorphic to a compact topological group. To see why a countably cellular Tychonoff space $X$ is not almost P, take a maximal disjoint family $\mathcal U$ of open $F_\sigma$-sets in $X$. Because of countable celularity this family is countable and hence $\bigcup \mathcal U$ is an open dense $F_\sigma$-subset and its complement is a nowhere dense closed $G_\delta$-set. $\endgroup$
    – Taras Banakh
    Commented May 19, 2019 at 18:56
  • $\begingroup$ A homogeneous compact almost $P$-space has a cover by closed nowhere dense $P$-sets. For examples of such compact (extremally disconnected) spaces, see this paper: arxiv.org/pdf/1809.05799.pdf $\endgroup$
    – Taras Banakh
    Commented May 19, 2019 at 18:58

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