Interior, Closure, and Boundary — Mathematics Knowledge System

Overview

Given a subset A of a topological space X, three companion sets describe how A sits inside X: the interior of A (the largest open set contained in A), the closure of A (the smallest closed set containing A), and the boundary of A (the points that are 'on the edge' — in the closure of both A and its complement). These three constructions recover, in any topological space, exactly the everyday geometric intuition of interior, closure, and edge for shapes in the plane, while also making sense for wild, non-metric spaces. Every point of X falls into exactly one of: the interior of A, the interior of the complement of A, or the boundary of A — a three-way partition of the whole space relative to A.

Intuition

Think of a filled-in disk in the plane that includes only part of its rim. The interior is everything strictly inside — points with a little open 'wiggle room' entirely inside the disk. The closure glues the whole rim back on, giving you the filled disk plus its full boundary circle, even the parts you started without. The boundary is exactly that rim: points where every neighborhood, no matter how small, catches some points inside the set and some points outside it. A point is interior if it has breathing room inside A; it is a boundary point if it can never fully escape A's presence or A's complement's presence, no matter how you shrink the neighborhood.

Formal Definition

Definition

Let (X, τ) be a topological space and A ⊆ X. Define:

\operatorname{int}(A) = \bigcup \{ U \in \tau : U \subseteq A \}

The union of all open subsets of A — the largest open set contained in A

Interior

\overline{A} = \bigcap \{ C \subseteq X : C \text{ closed}, A \subseteq C \}

The intersection of all closed supersets of A — the smallest closed set containing A

Closure

\partial A = \overline{A} \setminus \operatorname{int}(A) = \overline{A} \cap \overline{X \setminus A}

Points in the closure of A that are not interior to A; equivalently, points in the closure of both A and its complement

Boundary

X = \operatorname{int}(A) \sqcup \partial A \sqcup \operatorname{int}(X\setminus A)

Every point of X lies in exactly one of the interior of A, the boundary of A, or the interior of the complement of A

Trichotomy

Notation

Notation	Meaning
$\operatorname{int}(A) = A^{\circ}$	The interior of A
$\overline{A} = \operatorname{cl}(A)$	The closure of A
$\partial A$	The boundary (frontier) of A
$\overline{A} = X$	A is dense in X if its closure is the whole space — every nonempty open set of X meets A

Derivation

Deriving int(A) = X \ closure(X\A), the complementation identity linking interior and closure, directly from the definitions.

X\setminus \overline{X\setminus A} = X \setminus \bigcap\{C \text{ closed}: X\setminus A \subseteq C\}

Unfold the definition of closure applied to the complement of A

= \bigcup \{ X\setminus C : C \text{ closed}, X\setminus A \subseteq C \}

De Morgan: complement of an intersection is the union of complements

= \bigcup \{ U \in \tau : U \subseteq A \}

Substituting U = X\C (open, since C is closed) and noting X\A ⊆ C ⟺ U ⊆ A

= \operatorname{int}(A)

This union of open subsets of A is exactly the definition of the interior

Proofs

The boundary of A equals the closure of A intersected with the closure of its complement

$\partial A := \overline{A}\setminus \operatorname{int}(A)$ (Definition of boundary)
$\operatorname{int}(A) = X \setminus \overline{X\setminus A}$ (Complementation duality between interior and closure)
$\overline{A}\setminus \operatorname{int}(A) = \overline{A} \cap (X \setminus \operatorname{int}(A)) = \overline{A}\cap \overline{X\setminus A}$ (Substitute and simplify the set difference as an intersection with the complement)
$\therefore \partial A = \overline{A}\cap\overline{X\setminus A}$ (Combining the previous steps gives the symmetric closure-based formula for the boundary)

Properties

Interior is idempotent and open

\operatorname{int}(A) \in \tau, \qquad \operatorname{int}(\operatorname{int}(A)) = \operatorname{int}(A)

Closure is idempotent and closed

\overline{A} \text{ is closed}, \qquad \overline{\overline{A}} = \overline{A}

A is open iff A = int(A)

A \in \tau \iff A = \operatorname{int}(A)

A is closed iff A = closure(A)

A \text{ closed} \iff A = \overline{A}

Complementation duality

X \setminus \overline{A} = \operatorname{int}(X\setminus A), \qquad X \setminus \operatorname{int}(A) = \overline{X\setminus A}

Condition: Interior and closure are De Morgan duals of each other with respect to complementation.

Boundary of a boundary

\partial(\partial A) \subseteq \partial A

Example: For A = [0,1] ⊂ ℝ, ∂A = {0,1}, and ∂({0,1}) = {0,1} since {0,1} is already closed with empty interior.

Applications

Computational geometry algorithms (e.g. computing the convex hull or rasterizing a region) explicitly track a shape's boundary versus interior pixels to decide fill rules and edge-antialiasing.

Worked Examples

Every point of (0,1) has a small open interval around it still inside A, but 1 does not (any interval around 1 sticks out past 1). So the interior is (0,1).
$\operatorname{int}(A) = (0,1)$
The smallest closed set containing (0,1] must include the limit point 0 as well, giving [0,1].
$\overline{A} = [0,1]$
Boundary = closure minus interior: [0,1] \ (0,1) = {0,1}.
$\partial A = \{0,1\}$

Answer: int(A) = (0,1), closure(A) = [0,1], ∂A = {0,1}.

Practice Problems

Difficulty 4/10

The interior of a set A is defined as:

The smallest closed set containing AThe largest open set contained in AThe set of limit points of AA minus its boundary

Difficulty 5/10

Find the boundary of the set B = {(x,y) ∈ ℝ² : x² + y² < 1} (the open unit disk) in the standard topology on ℝ².

Difficulty 7/10

Prove that A is closed if and only if A contains its own boundary (∂A ⊆ A).

Quiz

Which formula correctly defines the boundary of A?

∂A = int(A) ∪ closure(A)∂A = closure(A) \ int(A)∂A = X \ A∂A = int(A) ∩ int(X\A)

If A is dense in X, which statement must be true?

int(A) = Xclosure(A) = X∂A = ∅A is open

Summary

int(A) is the largest open subset of A; closure(A) is the smallest closed superset of A; ∂A = closure(A) \ int(A) is the boundary.
Every point of X lies in exactly one of int(A), ∂A, or int(X\A) — a three-way partition determined by A.
Interior and closure are De Morgan duals: X\closure(A) = int(X\A), and X\int(A) = closure(X\A).
A is open iff A = int(A); A is closed iff A = closure(A) iff ∂A ⊆ A.
A is dense in X iff closure(A) = X iff every nonempty open subset of X meets A — as with ℚ in ℝ, where int(ℚ)=∅ yet closure(ℚ)=ℝ.

References

WebsiteWikipedia — Interior (topology)
WebsiteWikipedia — Closure (topology)
WebsiteWikipedia — Boundary (topology)
BookMunkres, J. Topology, 2nd ed. Ch. 2.