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A compact space $X$ is called ${\it \pi-monolithic}$ if whenever a surjective continuous mapping $f:X\rightarrow K$ where $K$ is a compact metric space there exists a compact metric space $T\subseteq X$ such that $f(T)=K$.

Perhaps this class of compact spaces has been studied (for example, in articles by R.Pol or T.A. Chapman). But I couldn't find a reference about this class of spaces.Therefore, for now we call this class as a class of $\pi$-monolithic compact spaces.

${\bf Question.}$ What is the name of the (possibly well-known) class of $\pi$-monolithic compact spaces?

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  • $\begingroup$ Do you have any non-trivial examples? $\endgroup$
    – KP Hart
    Commented Apr 2, 2022 at 7:38
  • $\begingroup$ For example, a monolithic compact space (or a dyadic compact space) is $\pi$-monolithic. $\endgroup$
    – Alexander Osipov
    Commented Apr 2, 2022 at 10:28
  • $\begingroup$ @AlexanderOsipov could you elaborate/give a reference on why monolithic compact spaces satisfy this property? (I'm only familiar with the definition of monolithicity in terms of cardinal characteristics) $\endgroup$
    – Alessandro Codenotti
    Commented Apr 2, 2022 at 16:07
  • $\begingroup$ Let $f:X->K$ be a continuous mapping $X$ onto a compact metric space $K$. Let $S$ be a countable dense subset of $K$. Then $T=\overline{f^{-1}(S)}$ is a compact metric space and $f(T)=K$. $\endgroup$
    – Alexander Osipov
    Commented Apr 3, 2022 at 5:12

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