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?
$A/R$ 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
- a discrete space is extremally disconnected;
- the Čech-Stone compactification of an extremally disconnected Tychonoff space is extremally disconnected and
- since $A$ is compact Hausdorff, $R$ is closed in $A \times A$ iff $A/R$ is Hausdorff.
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 $\beta X_d$.