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][1], 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][2] 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).


  [1]: https://mathoverflow.net/questions/202063/is-there-a-compact-connected-hausdorff-space-in-which-every-non-empty-g-delta
  [2]: https://www.cambridge.org/core/services/aop-cambridge-core/content/view/A5E6B7EE5B5F1CA2C16A30C75E468A4B/S0008414X00024597a.pdf/almostpspaces.pdf