Interiors, Closures and Neighbourhoods

Arun Ram
Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

and

Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 2 November 2009

Interiors and closures

Let $X$ be a topological space and let $x \in X$ . A neighbourhood of $x$ is a subset $N$ of $X$ such that there exists an open subset of $U$ of $X$ with $x \in U$ and $U \subseteq N$ .

Let $X$ be a topological space and let $E \subset X$ . A neighbourhood of $E$ is a subset $N$ of $X$ such that there exists an open subset $U$ of $X$ with $E \subseteq U \subseteq N$ .

Let $X$ be a topological space and let $E \subset X$ . The interior of $E$ is the subset $E^{o}$ of $E$ such that

$E^{o}$ is open in $X$
If $U$ is an open subset of $E$ then $U \subseteq E^{o}$ .

Let $X$ be a topological space and let $E \subset X$ . The closure of $E$ is the subset $\overline{E}$ of $E$ such that

$\overline{E}$ is closed,
If $V$ is a closed subset of $X$ and $V \supseteq E$ then $V \supseteq \overline{E}$ .

Let $X$ be a topological space and let $E \subseteq X$ . An interior point of $E$ is a point $x \in X$ such that there exists a neighbourhood $N_{x}$ of $x$ with $N_{x} \subseteq X$ .

Let $X$ be a topological space and let $E \subseteq X$ . An close point to $E$ is a point $x \in X$ such that if $N_{x}$ is a neighbourhood of $x$ then $N_{x}$ contains a point of $E$ .

Let

X

be a topological space. Let

E \subseteq X

The interior of $E$ is the set of interior points of $E$ .
The closure of $E$ is the set of close points of $E$ .

		Proof (of part a).
	Let $I = \{x \in E \| x is an interior point of E\}$ . To show that $E^{o} = I$ , we show that (aa) $I \subseteq E^{o}$ and then that (ab) $E^{o} \subseteq I$ . Let $x \in I$ . Then there exists a neighbourhood $N$ of $x$ with $N \subseteq E$ . So there exists an open set $U$ with $x \in U \subseteq N \subseteq E$ . Since $U \subseteq E$ and $U$ is open $U \subseteq E^{o}$ . So $x \subseteq E^{o}$ . So $I \subseteq E^{o}$ . We want to show that if $x \in E^{o}$ then $x \in I$ . Assume $x \in E^{o}$ . Then $E^{o}$ is open and $x \in E^{o} \subseteq E$ . So $x$ is an interior point of $E$ . So $x \in E^{o}$ . So $I \subseteq E^{o}$ .

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)