Question: does there exist an extremally disconnected infinite Hausdorff compact space $\ X\ $ such that the only homeomorphism $\ h: X\to X\ $ is the identity homeomorphism $\ h=\mathbb I_X:\ X\to X\quad $?
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1$\begingroup$ Sir, probably you already possess this knowledge : $I_X$ is the only homeomorphism iff the identity map is the only order isomorphism on the boolean algebra of the clopen subsets of $X$. $\endgroup$– Onur OktayCommented Aug 16, 2023 at 0:32
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$\begingroup$ Dear @OnurOktay, it's been more than half a century ago when I was into extremely disconnected spaces and related question, by now I am rusty. $\endgroup$– Wlod AACommented Aug 16, 2023 at 1:12
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3$\begingroup$ You might look at this question mathoverflow.net/questions/309122/… $\endgroup$– Benjamin SteinbergCommented Aug 16, 2023 at 1:12
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$\begingroup$ @BenjaminSteinberg, indeed, very interesting, thank you! $\endgroup$– Wlod AACommented Aug 16, 2023 at 1:20
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1 Answer
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This paper, Rigid Stone spaces within $\mathsf{ZFC}$ by Dow, Gubbi, and Szymański contains relatively easy examples of such spaces.
