## Interiors and closures

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

(a)   ${E}^{o}$ is open and ${E}^{o}\subseteq E$, and
(b)   if $U$ is open and $U\subseteq E$ then $U\subseteq {E}^{o}$.
The closure of $E$ is the subset $\stackrel{‾}{E}$ of $X$ such that
(a)   $\stackrel{‾}{E}$ is closed and $\stackrel{‾}{E}\supseteq E$, and
(b)   if $V$ is closed and $V\supseteq E$ then $V\supseteq \stackrel{‾}{E}$.

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 $N$ of $x$ such that $N\subseteq E$.
A close point of $E$ is a point $x\in X$ such that if $N$ is a neighbourhood of $x$ then $N\cap E\ne \varnothing$.

Let $X$ be a topological space. Let $E\subseteq X$.

(a)   The interior of $E$ is the set of interior points of $E$.
(b)   The closure of $E$ is the set of close points of $E$.

 Proof (of part a). Let $I=\left\{x\in E\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}x\phantom{\rule{0.5em}{0ex}}\text{is an interior point of}\phantom{\rule{0.5em}{0ex}}E\right\}.$ 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 $I\subseteq {E}^{o}$.

HW: Let $X$ be a topological space and let $E\subseteq X\text{.}$

 (a) Show that $\stackrel{‾}{{E}^{c}}={\left({E}^{\circ }\right)}^{c}$ by using the definition of closure. (b) Show that ${\left({E}^{c}\right)}^{\circ }={\left(\stackrel{‾}{E}\right)}^{c},$ by taking complements and using (a). (c) Show that $C=\left\{x\in X | \text{if} N\in 𝒩\left(x\right) \text{then} N\cap E\ne \varnothing \right\}$ is the set of close points of $E\text{.}$ (d) Show that ${C}^{c}={\left({E}^{c}\right)}^{\circ }\text{.}$ (e) Show that $C=\stackrel{‾}{E}\text{.}$

Proof.

(a)
To show: (aa) ${\left({E}^{\circ }\right)}^{c}$ is closed and ${\left({E}^{\circ }\right)}^{c}\supseteq {E}^{c}\text{.}$
(ab) If $V$ is closed and $V\supseteq {E}^{c}$ then $V\supseteq {\left({E}^{\circ }\right)}^{c}\text{.}$
 (aa) Since ${E}^{\circ }$ is open, then ${\left({E}^{\circ }\right)}^{c}$ is closed. Since ${E}^{\circ }\subseteq E$ then ${\left({E}^{\circ }\right)}^{c}\supseteq {E}^{c}\text{.}$ (ab) Assume $V$ is closed and $V\supseteq {E}^{c}\text{.}$ Then ${V}^{c}$ is open and ${V}^{c}\subseteq E\text{.}$ So ${V}^{c}\subseteq {E}^{\circ }\text{.}$ So $V\supseteq {\left({E}^{\circ }\right)}^{c}\text{.}$
So ${\left({E}^{\circ }\right)}^{c}=\stackrel{‾}{{E}^{c}}\text{.}$
(b)
 To show: ${\left({E}^{c}\right)}^{\circ }={\left(\stackrel{‾}{E}\right)}^{c}\text{.}$ To show: ${\left({\left({E}^{c}\right)}^{\circ }\right)}^{c}=\stackrel{‾}{E}\text{.}$ By (a), ${\left({\left({E}^{c}\right)}^{\circ }\right)}^{c}=\stackrel{‾}{{\left({E}^{c}\right)}^{c}}=\stackrel{‾}{E}\text{.}$
(c) By definition of close point $C=\left\{x\in X | \text{if} N\in 𝒩\left(x\right) \text{then} N\cap E\ne \varnothing \right\}$ is the set of close points of $E\text{.}$
(d) By definition of $C,$ $Cc = { x∈X | there exists N∈𝒩(x) such that N∩E=∅ } = { x∈X | there exists N∈𝒩(x) such that N⊆Ec } ,$ which is the set of interior points of ${E}^{c}\text{.}$ Thus ${C}^{c}={\left({E}^{c}\right)}^{\circ },$ by Proposition 1.1(a).
(e)
 To show: $C=\stackrel{‾}{E}\text{.}$ To show: ${C}^{c}={\left(\stackrel{‾}{E}\right)}^{c}\text{.}$ By (d) and (b), ${C}^{c}={\left({E}^{c}\right)}^{\circ }={\left(\stackrel{‾}{E}\right)}^{c}\text{.}$

$\square$

## Notes and References

These notes follow Bourbaki [Bou, Ch. 1 § 1.6].

The definition of the interior of $E$ is the mathematically precise formulation of $\text{"}{E}^{\circ }$ is the largest open set contained in $E\text{".}$ The definition of the closure of $E$ is the mathematically precise formulation of $\text{"}\stackrel{‾}{E}$ is the smallest closed sets containing $E\text{".}$ These notes follow Bourbaki [Bou, Ch. I §1 no. 6]. Similar information is treated in [Ru, Ch. 2, 2.18-2.27].

## References

[Bou] N. Bourbaki, General Topology, Springer-Verlag, 1989. MR1726779.

[BR] W. Rudin, Principles of mathematical analysis, Third edition, International Series in Pure and Applied Mathematics, McGraw-Hill 1976. MR0385023.

[Ru] W. Rudin, Real and complex analysis, Third edition, McGraw-Hill, 1987. MR0924157.