$\DeclareMathOperator\cl{cl}\DeclareMathOperator\int{int}$A subset $A$ of a topological space $X$ is called regular closed if $A=\cl _{X}\int_{X}A$.

The family of all regular closed sets of a topological space is denoted by $% \mathcal{R}\left( X\right) $.

An ultrafilter $\mathcal{U}$ on $\mathcal{R}\left( X\right) $ is said to converge to a point $p\in \beta X$ if $\left\{ p\right\} =\bigcap \left\{ \cl_{\beta X}U:U\in \mathcal{U}\right\} $.

Lemma: Let $D$ be a dense subspace of a space $X$. Then the map $% A\rightarrow \cl_{X}A$ is a Boolean algebra isomorphism from $\mathcal{R}% \left( D\right) $ onto $\mathcal{R}\left( X\right) $.

I think that the family $\mathcal{F}=\left\{ F\in \mathcal{R}\left( \beta X\right) :p\in \int_{\beta X}F\right\} $ is a filterbasis in $\mathcal{R}% \left( \beta X\right) $. Therefore $\mathcal{F}$ can be imbedded in an ultrafilter $\mathcal{U}$ in $\mathcal{R}\left( \beta X\right) $. Therefore $% X\cap \mathcal{U}=\left\{ X\cap U:U\in \mathcal{U}\right\} $ is an ultrafilter in $\mathcal{R}\left( X\right) $, and it converges to $p$.

My question is: for every $p\in \beta X$, does there exist a unique ultrafilter $% \mathcal{U}$ in $\mathcal{R}\left( X\right) $ such that $\left\{ p\right\} =\bigcap \left\{ cl_{\beta X}U:U\in \mathcal{U}\right\} $, that is $% \mathcal{U}$ converges to $p$?

Now, let $f:X\longrightarrow Y$ be a continuous map between Tychonoff spaces. Then the Stone extension $\beta f:\beta X\longrightarrow \beta Y$ is defined as follows: for $p\in \beta X$, there exists a unique $z$-ultrafilter $\mathcal{% A}^{p}$ on $X$ with $p$, so is defined by $\left( \beta f\right) \left( p\right) =\bigcap f^{\#}\mathcal{A}^{p}$, where $f^{\#}\mathcal{A}% ^{p}=\left\{ E\in Z\left( Y\right) :f^{-1}\left( E\right) \in \mathcal{A}% ^{p}\right\} $ (Gillman and Jerison, Rings of continuous functions, p.85). In another article (K. Srivastava. On the Stone–Čech compactification of an orbit space), it is defined by $\left( \beta f\right) \left( p\right) =\bigcap_{Z\in \mathcal{A}^{p}}\cl_{\beta Y}f\left( Z\right) $. I guess that's not quite right.

If, for every $p\in \beta X$, there exists a unique ultrafilter of regular closed sets of $X$, then how can I define $\left( \beta f\right) \left( p\right) $?

`\left`

and`\right`

ing all pairs of parentheses is harmless in terms of sizing if the contents are small enough (though it can be really weird looking when the contents are asymmetric about the midline: $\displaystyle\left(\sum_n n^{-s}\right)$), but it does weird things to the spacing. Compare $\mathcal R(X)$`\mathcal R(X)`

vs. $\mathcal R\left(X\right)$`\mathcal R\left(X\right)`

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