# A variant of the Moore-Mrowka problem

• A space $$X$$ is said to be sequential if whenever $$A \subset X$$ is not closed then $$A$$ contains a sequence converging to a point outside of $$A$$.
• A space $$X$$ is said to have countable tightness if for every non-closed set $$A \subset X$$ and every point $$x \in \overline{A} \setminus A$$ there is a countable subset $$C$$ of $$A$$ such that $$x \in \overline{C}$$.

Clearly, every sequential space has countable tightness, and not every space of countable tightness is sequential. The Moore-Mrowka problem asks whether every compact Hausdorff space of countable tightness is sequential.

This problem is known to independent of ZFC. Balogh proved that every compact space of countable tightness is sequential under PFA and the one-point compactification of Ostaszewski's example under $$\Diamond$$ of a locally compact countably compact hereditarily separable perfectly normal non-compact space, is an example of a compact space of countable tightness which is not sequential.

Balogh, Zoltán, On compact Hausdorff spaces of countable tightness, Proc. Am. Math. Soc. 105, No. 3, 755-764 (1989). ZBL0687.54006.

Even though it is not sequential, the one-point compactification of Ostaszewski space enjoys another natural convergence-type property known as pseudoradiality. A space is pseudoradial if whenever $$A \subset X$$ is not closed then $$A$$ contains a transfinite sequence (that is, a well-ordered net) converging outside of $$A$$.

$$MA_{\omega_1}$$ negates the existence of Ostaszewski space for two reasons: 1) because it implies that every countably compact perfectly normal space is compact (Weiss) or 2) because it implies that every locally compact hereditarily separable space is hereditarily Lindelof (Szentmiklossy). However, $$MA_{\omega_1}$$ is not strong enough to imply a positive solution to the Moore-Mrowka problem (see Balogh's paper).

QUESTION: Assume $$MA_{\omega_1}$$. Is it true then that every compact pseudoradial space of countable tightness is sequential?

NOTE: 1) A byproduct of Szentmiklossy's result is that the Moore-Mrowka problem has a positive answer for hereditarily separable spaces under $$MA_{\omega_1}$$ (every hereditarily Lindelof space has points $$G_\delta$$ and every compact space with points $$G_\delta$$ is first-countable, and hence, sequential).

2) A Hausdorff (non-compact) pseudoradial non-sequential space of countable tightness was constructed in ZFC by Simon and Tironi.

Simon, Petr; Tironi, Gino, Two examples of pseudo-radial spaces, Commentat. Math. Univ. Carol. 27, 155-161 (1986). ZBL0596.54005.