Interior, Exterior, Boundary, Closure

Introduction

In our previous post, we introduced the foundational concept of a topological space. Here, we advance our study by formalizing four cornerstone notions—interior, closure, exterior, and boundary—entirely in terms of open sets, without recourse to any metric. Though these concepts often arise in metric spaces, their true generality and elegance emerge in the broader context of topology.

Today’s goal is to present concise, precise definitions and to illustrate how these constructs partition a topological space into regions of “definitive membership,” “indefinite membership,” and “definitive non‑membership.”

The Interior

Let $(X,\tau)$ be a topological space and $A\subset X$ . The interior of $A$ , denoted $\operatorname{Int}(A)$ or $A^o$ , is defined to be, $\operatorname{int}(A)\;=\;\bigcup\{\,U\in\tau:U\subseteq A\}$ .

Equivalently, $\operatorname{int}(A)$ is the unique largest open set contained in $A$ . In other words, $U\subset A,U\in\tau\implies U\subset A^o$ .

A point $x\in A$ is said to be an interior point of $A$ if and only if $x\in\operatorname{int}(A)$ . Equivalently, $x\in A$ is an interior point of $A$ if and only if there exists an open set $U\subset\tau$ with $x\in U\subset A$ .

Example 1.2.
In the real line $\mathbb R$ with its standard topology, $\operatorname{int}([0,1])=(0,1)$ .

The endpoints $0$ and $1$ are excluded, for no open set of $\mathbb R$ around them can lie entirely within $[0,1]$ .

The Closure

The closure of $A$ , denoted $\overline A$ , is defined by:

$\overline{A}\;=\;\bigcap\{\,F\subseteq X:F\text{\space is closed and\space}A\subseteq F\}$ .

Equivalently, $\overline A$ is the unique smallest closed set containing $A$ .

A point $x\in X$ belongs to $\overline A$ if and only if every open neighborhood $U\ni x$ intersects $A$ . We call the elements of $\overline A$ as the adherent points of $A$ .

Example: The closure of the open interval $(0,1)$ in $\mathbb R$ is the closed interval $[0,1]$ , since any open set of $\mathbb R$ containing $0$ or $1$ intersects $(0,1)$ , showing these endpoints are also adherent points of $(0,1)$ .

The Exterior

The exterior of a set $A$ , denoted $\operatorname{Ext}(A)$ , is simply the interior of the complement of $A$ , i.e., $\operatorname{Int}(X\setminus A)$ .

Thus $\operatorname{Ext}(A)$ is the largest open set disjoint from $A$ .

Equivalently, $x\in\operatorname{Ext}(A)$ if and only if there exists an open set $U\ni x$ with $U\cap A=\varnothing$ .

Example:
Again in $\mathbb R$ , for $A=[0,1]$ , the exterior is $(-\infty,0)\cup(1,\infty)$ .

The Boundary

The boundary of a set $A$ , denoted $\partial A$ is defined as $\partial A=\overline{A}\setminus\operatorname{int}(A)$ .

Equivalently, $\partial A$ consists of those points whose every open neighborhood meets both $A$ and its complement $(X\setminus A)$ .

The boundary is where things get interesting. It’s the set of points that neither belong to the interior nor to the exterior.

Example: For $A=(0,1)$ , the boundary is $\partial A=\{0,1\}$ . For $A=\mathbb Q$ , the set of rationals in $\mathbb R$ , the boundary is $\partial A=\mathbb R$ , because rationals are dense and so are irrationals!

Fundamental Relationships

These four constructs satisfy elegant identities that clarify how they carve up $X$ :

$\operatorname{Int}(A)\cup\partial A\cup\operatorname{Ext}(A)=X$
$\overline{A}=\operatorname{int}(A)\cup\partial A$
$\partial A=\overline A\cap\overline{(X\setminus A)}$
$\partial(X\setminus A)=\partial A$

These identities underscore that every point of $X$ either lies strictly inside $A$ , strictly outside $A$ , or precisely on the edge where membership is tenuous.

Concluding Remarks

The quartet of interior, closure, exterior, and boundary provides a topological atlas for navigating any subset. Beyond metric intuitions of “distance,” these notions capture the essence of openness, closedness, and adjacency purely through the language of neighborhoods.

As you further explore continuous maps, connectedness, and compactness, you will find these four ideas recur repeatedly—illuminating the subtle “shape” of spaces where geometry yields to topology.

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