| Version current |
Version 15 |
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| Let $A$ be a subset of a topological space $X$. |
Let $A$ be a subset of a topological space $X$. |
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| The union of all open sets contained in $A$ |
The union of all open sets contained in $A$ |
| is defined to be the \emph{interior} of $A$. |
is defined to be the \emph{interior} of $A$. |
| Equivalently, one could define the interior |
Equivalently, one could define the interior |
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We denote the interior of $A$ by $\int(A)$. |
| In this entry we denote the interior of $A$ by $\int(A)$. |
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| Another common notation is $A^\circ$. |
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| The \emph{exterior} of $A$ is defined as |
The \emph{exterior} of $A$ is defined as |
| the union of all open sets whose intersection with $A$ is empty. |
the union of all open sets whose intersection with $A$ is empty. |
| That is, the exterior of $A$ is the interior of the complement of $A$. |
That is, the exterior of $A$ is the interior of the complement of $A$. |
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| The interior of a set enjoys many special properties, |
The interior of a set enjoys many special properties, |
| some of which are listed below: |
some of which are listed below: |
| \begin{enumerate} |
\begin{enumerate} |
| \item $\int(A)\subseteq A$ |
\item $\int(A)\subseteq A$ |
| \item $\int(A)$ is open |
\item $\int(A)$ is open |
| \item $\int(\int(A))=\int(A)$ |
\item $\int(\int(A))=\int(A)$ |
| \item $\int(X)=X$ |
\item $\int(X)=X$ |
| \item $\int(\emptyset)=\emptyset$ |
\item $\int(\emptyset)=\emptyset$ |
| \item $A$ is open if and only if $A=\int(A)$ |
\item $A$ is open if and only if $A=\int(A)$ |
| \item $\overline{A^\complement}=(\int(A))^\complement$ |
\item $\overline{A^\complement}=(\int(A))^\complement$ |
| \item $\overline{A}^\complement = \int(A^\complement)$ |
\item $\overline{A}^\complement = \int(A^\complement)$ |
| \item $A\subseteq B$ implies that $\int(A)\subseteq \int(B)$ |
\item $A\subseteq B$ implies that $\int(A)\subseteq \int(B)$ |
| \item $\int(A)=A\setminus \partial A$, |
\item $\int(A)=A\setminus \partial A$, |
| where $\partial A$ is the boundary of $A$ |
where $\partial A$ is the boundary of $A$ |
| \item $X=\int(A)\cup \partial A \cup \int(A^\complement)$ |
\item $X=\int(A)\cup \partial A \cup \int(A^\complement)$ |
| \end{enumerate} |
\end{enumerate} |
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| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{willard} S. Willard, \emph{General Topology}, |
\bibitem{willard} S. Willard, \emph{General Topology}, |
| Addison-Wesley Publishing Company, 1970. |
Addison-Wesley Publishing Company, 1970. |
| \end{thebibliography} |
\end{thebibliography} |
