= Gleason cover of a compact regular locale
{c}
{title2=$\gamma X$}
For a compact <regular locale> $X$, put $B=\mathcal O(X)_{\neg\neg}$ and $\mathcal O(\gamma X)=\operatorname{Idl}(B)$. This <locale> is extremally disconnected. Its natural surjection to $X$ has inverse image
$$
a\longmapsto\bigvee_{u\prec a}\mathord\downarrow(\neg\neg u).
$$
The <Joyal extension lemma> gives the <frame homomorphism>; its injectivity follows by recovering $a$ as the join of the regular opens in this ideal.
Back to article page
