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Let $A$ be a compact Hausdorff space which is also extremally disconnected (meaning that the closure of any open subset is still open). Suppose also that $R$ is an equivalence relation on $A$ which is closed in the product topology $A\times A$. Is it the case that the quotient space $A/R$ is a compact Hausdorff and extremally disconnected space?

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$A/R$ is a compact Hausdorff space, but nothing more can be said about it, since every compact Hausdorff space is a continuous image, and therefore homeomorphic to a quotient of an extremally disconnected compact Hausdorff space. This is easy to see if you know that

  1. a discrete space is extremally disconnected;
  2. the Čech-Stone compactification of an extremally disconnected Tychonoff space is extremally disconnected and
  3. since $A$ is compact Hausdorff, $R$ is closed in $A \times A$ iff $A/R$ is Hausdorff.
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Let $X$ be a compact Hausdorff space and consider $X_d$ as a set (that is $X$ endowed with the discrete topology). The inclusion $\iota\colon X_d\to X$ is continuous. Therefore, it extends to the (unique) continuous map $\beta\iota\colon \beta X_d\to X$. Note that $\beta \iota$ is surjective therefore $X$ is a quotient of the extremely disconnected space $\beta X_d$.

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