Let $X$ be a compact Hausdorff space and let $e(X)$ be it's projective cover, i.e. the Stone space of the Boolean algebra of the regular closed subsets of $X$. Let $e:e(X)\rightarrow X$, $u\mapsto\cap u$ be the map from Gleason's theorem. $e$ is an irreducible surjection. My question is the following:
Let's concider the map $A\mapsto$ cl(int($A$)). Is it true that this is a bijection between the regular closed subsets of $X$ and the clopen subsets of $e(X)$?
It is obvious that the image of each regular closed set is a clopen set in $e(X)$
I'm having troubles proving that it maps different RC sets to different clopen sets.
I tried starting with two different RC sets $A\neq B$ and taking, without loss of generality, that $x\in A$\ $B$. I couldn't do anything productive, so I then assumed that $x\in$ int($A$)\int($B$), because two different regular closed sets must have different interiors.I couldn't get that to work too.