Some observations on compact indestructible spaces (Q387189)

A compact (resp. Lindelöf) space is \textit{indestructible} if it remains compact (resp. Lindelöf) in any countably closed forcing extension. In this note, the author studies compact or Lindelöf indestructible spaces. Among other things, it is shown that a compact indestructible space is sequentially compact, a Lindelöf \(T_2\) indestructible space has the finite derived set property, and a compact \(T_2\) indestructible space is pseudoradial. The last fact sharpens a recent result of \textit{R. R. Dias} and \textit{F. D. Tall} in [Topology Appl. 160, No. 18, 2411--2426 (2013; Zbl 1295.54026)]. It is also observed that under CH, a compact weakly Whyburn space of countable tightness is indestructible. Several interesting open questions relevant to indestructible spaces are posed.