$\DeclareMathOperator\cl{cl}\DeclareMathOperator\int{int}$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](https://doi.org/10.1007/978-1-4615-7819-2), p.85). In another article (K. Srivastava. [On the Stone–Čech compactification of an
orbit space](https://doi.org/10.1017/S0004972700003725)), 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) $?