| Version 4 |
Version 3 |
| Let $X$ be a topological space. A subset $A$ of $X$ is called a |
Let $X$ be a topological space. A subset $A$ of $X$ is called a |
| \emph{regular open} set if |
\emph{regular open} set if |
| \begin{enumerate} |
\begin{enumerate} |
| \item $A$ is open, and |
\item $A$ is open, and |
| \item $A$ is equal to the interior of the closure of itself: |
\item $A$ is equal to the interior of the closure of itself: |
| $A=\operatorname{int}(\overline{A})$. |
$A=\operatorname{int}(\overline{A})$. |
| \end{enumerate} |
\end{enumerate} |
| \textbf{Examples}. Let $\mathbb{R}$ be the real line with the usual |
\textbf{Examples}. Let $\mathbb{R}$ be the real line with the usual |
| topology (generated by open intervals). |
topology (generated by open intervals). |
| \begin{itemize} |
\begin{itemize} |
| \item $(a,b)$ is regular open whenever $-\infty<a\leq b<\infty$. |
\item $(a,b)$ is regular open whenever $-\infty<a\leq b<\infty$. |
| \item $(a,b)\cup(b,c)$ is not regular open for $-\infty<a\leq b |
\item $(a,b)\cup(b,c)$ is not regular open for $-\infty<a\leq b |
| \leq c<\infty$ and $a\neq c$. The interior of the closure of |
\leq c<\infty$ and $a\neq c$. The interior of the closure of |
| $(a,b)\cup(b,c)$ is $(a,c)$. |
$(a,b)\cup(b,c)$ is $(a,c)$. |
| \end{itemize} |
\end{itemize} |
| If we examine the structure of $\operatorname{int}(\overline{A})$ a |
If we examine the structure of $\operatorname{int}(\overline{A})$ a |
| little more closely, we see that if we define |
little more closely, we see that if we define |
|
$$A^{\bot}:=X-\overline{A},$$ then $$A^{\bot\bot}=
|
$A^{\bot}:=X-\overline{A},$ then $$A^{\bot\bot}=
|
| \operatorname{int}(\overline{A}).$$ So an alternative definition of |
\operatorname{int}(\overline{A}).$$ So an alternative definition of |
| a regular open set is an open set $A$ such that $A^{\bot\bot}=A$. |
a regular open set is an open set $A$ such that $A^{\bot\bot}=A$. |
|
|
| \textbf{Remarks}. |
\textbf{Remarks}. |
| \begin{itemize} |
\begin{itemize} |
| \item For any $A\subseteq X$, $A^{\bot}$ is always open. |
\item For any $A\subseteq X$, $A^{\bot}$ is always open. |
| \item $\varnothing^{\bot}=X$ and $X^{\bot}=\varnothing$. |
\item $\varnothing^{\bot}=X$ and $X^{\bot}=\varnothing$. |
| \item $A\cap A^{\bot}=\varnothing$ and $A\cup A^{\bot}$ is dense in |
\item $A\cap A^{\bot}=\varnothing$ and $A\cup A^{\bot}$ is dense in |
| $X$. |
$X$. |
| \item $A^{\bot}\cup B^{\bot}\subseteq(A\cap |
\item $A^{\bot}\cup B^{\bot}\subseteq(A\cap |
| B)^{\bot}$ and $A^{\bot}\cap B^{\bot}=(A\cup B)^{\bot}$. |
B)^{\bot}$ and $A^{\bot}\cap B^{\bot}=(A\cup B)^{\bot}$. |
| \item It can be shown that if $A$ is open, then $A^{\bot}$ is |
\item It can be shown that if $A$ is open, then $A^{\bot}$ is |
| regular open. As a result, following from the first property, $\operatorname{int}(\overline{A})$, being $A^{\bot\bot}$, is regular open for any subset $A$ of $X$. |
regular open. As a result, following from the first property, $\operatorname{int}(\overline{A})$, being $A^{\bot\bot}$, is regular open for any subset $A$ of $X$. |
| \item In addition, if both $A$ and $B$ are regular open, then $A\cap |
\item In addition, if both $A$ and $B$ are regular open, then $A\cap |
| B$ is regular open. |
B$ is regular open. |
| \item It is not true, however, that the union of two regular open |
\item It is not true, however, that the union of two regular open |
| sets is regular open, as illustrated by the second example above. |
sets is regular open, as illustrated by the second example above. |
| \item It can also be shown that the set of all regular open sets of |
\item It can also be shown that the set of all regular open sets of |
| a topological space $X$ forms a Boolean algebra under the following |
a topological space $X$ forms a Boolean algebra under the following |
| set of operations: |
set of operations: |
| \begin{enumerate} |
\begin{enumerate} |
| \item $1=X$ and $0=\varnothing$, |
\item $1=X$ and $0=\varnothing$, |
| \item $a\land b=a\cap b$, |
\item $a\land b=a\cap b$, |
| \item $a\lor b=(a\cup b)^{\bot\bot}$, and |
\item $a\lor b=(a\cup b)^{\bot\bot}$, and |
| \item $a^{\prime}=a^{\bot}$. |
\item $a^{\prime}=a^{\bot}$. |
| \end{enumerate} |
\end{enumerate} |
| This is an example of a Boolean algebra coming from a collection of |
This is an example of a Boolean algebra coming from a collection of |
| subsets of a set that is not formed by the standard set operations |
subsets of a set that is not formed by the standard set operations |
| union $\cup$, intersection $\cap$, and complementation $^{\prime}$. |
union $\cup$, intersection $\cap$, and complementation $^{\prime}$. |
| \end{itemize} |
\end{itemize} |
|
|
| The definition of a regular open set can be dualized. A closed set $A$ in a topological space is called a \emph{regular closed set} if $A=\overline{\operatorname{int}(A)}$. |
The definition of a regular open set can be dualized. A closed set $A$ in a topological space is called a \emph{regular closed set} if $A=\overline{\operatorname{int}(A)}$. |
|
|
| \begin{thebibliography}{[AHU]} |
\begin{thebibliography}{[AHU]} |
| \bibitem{halmos} P. Halmos (1970). {\em{Lectures on Boolean Algebras}}, Springer. |
\bibitem{halmos} P. Halmos (1970). {\em{Lectures on Boolean Algebras}}, Springer. |
| \bibitem{willard} S. Willard (1970). \emph{General Topology}, Addison-Wesley Publishing Company. |
\bibitem{willard} S. Willard (1970). \emph{General Topology}, Addison-Wesley Publishing Company. |
| \end{thebibliography} |
\end{thebibliography} |
