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LSpice
LSpice
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$\DeclareMathOperator\cl{cl}\DeclareMathOperator\int{int}$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) $$\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) $?

$\DeclareMathOperator\cl{cl}\DeclareMathOperator\int{int}$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) $?

$\DeclareMathOperator\cl{cl}\DeclareMathOperator\int{int}$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) $?

Proofreading and links to articles
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LSpice
LSpice
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Stone-Čech Stone–Čech compactification and aan ultrafilter of regular closed sets

A$\DeclareMathOperator\cl{cl}\DeclareMathOperator\int{int}$A subset $A$ of a topological space $X$ is called regular closed if $A=cl _{X}Int_{X}A$$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\} $$\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$$% 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\} $$\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 aan 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 aan ultrafilter in $\mathcal{R}\left( X\right) $, and it converges to $p$.

My question is: for every $p\in \beta X$, does there existsexist 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 continuoscontinuous 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 continuos functionsRings of continuous functions, p.85). If inIn another article (K. Srivastava. On the Stone-Cech compactification of an orbit spaceOn 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 iI define $\left( \beta f\right) \left( p\right) $?

Stone-Čech compactification and a ultrafilter of regular closed sets

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 a 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 a ultrafilter in $\mathcal{R}\left( X\right) $, and it converges to $p$.

My question for every $p\in \beta X$, there exists 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 continuos map between Tychonoff spaces. Then 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 continuos functions, p.85). If in another article (K. Srivastava. On the Stone-Cech 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) $?

Stone–Čech compactification and an ultrafilter of regular closed sets

$\DeclareMathOperator\cl{cl}\DeclareMathOperator\int{int}$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) $?

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Mehmet Onat
Mehmet Onat
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Stone-Čech compactification and a ultrafilter of regular closed sets

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 a 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 a ultrafilter in $\mathcal{R}\left( X\right) $, and it converges to $p$.

My question for every $p\in \beta X$, there exists 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 continuos map between Tychonoff spaces. Then 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 continuos functions, p.85). If in another article (K. Srivastava. On the Stone-Cech 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) $?