Interiors and closures

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

Last updates: 4 March 2014

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

\overline{E}

X

such that

(a)

\overline{E}

is closed and

\overline{E} \supseteq E

, and

(b) if

V

is closed 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 $N$ 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 $N \cap E \neq \emptyset$ .

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 $I = {x \in E \| x is an interior point of E} .$ 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 $I \subseteq E^{o}$ .

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

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

Proof.

(a)

To show:

(aa)

{(E^{\circ})}^{c}

is closed and

{(E^{\circ})}^{c} \supseteq E^{c} .

(ab)

V

is closed and

V \supseteq E^{c}

then

V \supseteq {(E^{\circ})}^{c} .

(aa)	Since $E^{\circ}$ is open, then ${(E^{\circ})}^{c}$ is closed. Since $E^{\circ} \subseteq E$ then ${(E^{\circ})}^{c} \supseteq E^{c} .$
(ab)	Assume $V$ is closed and $V \supseteq E^{c} .$ Then $V^{c}$ is open and $V^{c} \subseteq E .$ So $V^{c} \subseteq E^{\circ} .$ So $V \supseteq {(E^{\circ})}^{c} .$

{(E^{\circ})}^{c} = \overline{E^{c}} .

(b)

To show:	${(E^{c})}^{\circ} = {(\overline{E})}^{c} .$
To show:	${({(E^{c})}^{\circ})}^{c} = \overline{E} .$
By (a), ${({(E^{c})}^{\circ})}^{c} = \overline{{(E^{c})}^{c}} = \overline{E} .$

(c)

By definition of close point

C = {x \in X | if N \in 𝒩 (x) then N \cap E \neq \emptyset}

is the set of close points of

E .

(d)

By definition of

C,

\begin{matrix} C^{c} & = & {x \in X | there exists N \in 𝒩 (x) such that N \cap E = \emptyset} \\ = & {x \in X | there exists N \in 𝒩 (x) such that N \subseteq E^{c}}, \end{matrix}

which is the set of interior points of

E^{c} .

Thus

C^{c} = {(E^{c})}^{\circ},

by Proposition 1.1(a).

(e)

To show:	$C = \overline{E} .$
To show:	$C^{c} = {(\overline{E})}^{c} .$
By (d) and (b), $C^{c} = {(E^{c})}^{\circ} = {(\overline{E})}^{c} .$

$□$

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 $" E^{\circ}$ is the largest open set contained in $E ".$ The definition of the closure of $E$ is the mathematically precise formulation of $" \overline{E}$ is the smallest closed sets containing $E ".$ 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.

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