Open and Closed Sets — Mathematics Knowledge System

Overview

In a topological space (X, τ), the open sets are by definition exactly the members of τ, and a closed set is defined as a set whose complement in X is open. Open and closed sets are generalizations of the open interval (a,b) and closed interval [a,b] on the real line — but unlike common English usage of 'open' and 'closed' as opposites, a set can be both open and closed (clopen) or neither. Open sets formalize 'having wiggle room around every point'; closed sets formalize 'containing all of their own limit points.' Together they are the basic vocabulary from which continuity, closure, boundary, and compactness are all built.

Intuition

An open set is one where every point has some breathing room — you can wiggle a point in any direction by a small enough amount and stay inside the set. The open interval (0,1) is open: no matter how close x is to 0, there's a smaller interval around x still inside (0,1). A closed set, by contrast, is one that includes all the points it's 'approaching' — its boundary belongs to the set. [0,1] is closed: it contains its edge points 0 and 1, so no sequence of points inside [0,1] can sneak up to a limit outside the set. Crucially, open and closed are not opposites: [0,1) contains one boundary point (0) but not the other (1), so it is neither open nor closed.

Formal Definition

Definition

Given a topological space (X, τ), a set U ⊆ X is open if U ∈ τ. A set C ⊆ X is closed if its complement is open:

C \text{ is closed} \iff X \setminus C \text{ is open} \iff X\setminus C \in \tau

Definition of a closed set

I = (0,1) \text{ is open in } \mathbb{R}; \quad I^{\complement} = (-\infty, 0] \cup [1, \infty) \text{ is its (closed) complement}

The open interval and its closed complement, illustrating the defining duality

J = [0,1] \text{ is closed}; \quad K = [0,1) \text{ is neither open nor closed}

K fails to be open (no interval around 0 stays inside K without also needing room below 0... more precisely, no wiggle room exists at points approaching 1 from inside K staying in K) and fails to be closed since its complement ℝ\K=(-∞,0)∪[1,∞) is not open (no room around the point 1)

Notation

Notation	Meaning
$X \setminus C \text{ or } C^{\complement}$	The complement of C within the ambient space X
$\overline{A} = \operatorname{cl}(A)$	The closure of A — the smallest closed set containing A, equivalently A together with all its limit points
$\operatorname{int}(A) = A^{\circ}$	The interior of A — the largest open set contained in A
$\partial A = \overline{A} \setminus A^{\circ}$	The boundary of A — points in the closure but not in the interior
$\text{clopen}$	A set that is simultaneously open and closed — always true of ∅ and X in any topological space

Derivation

Deriving that finite unions of closed sets are closed, from the open-set axiom that finite intersections of open sets are open, via De Morgan's law.

C_1, \ldots, C_n \text{ closed} \implies X\setminus C_1, \ldots, X\setminus C_n \text{ open}

By definition of closed

\bigcap_{i=1}^{n}(X\setminus C_i) \text{ is open}

Finite intersection of open sets is open, by the topology axioms

X \setminus \bigcup_{i=1}^{n} C_i = \bigcap_{i=1}^{n}(X\setminus C_i)

De Morgan's law: the complement of a union is the intersection of complements

\therefore X\setminus\bigcup_{i=1}^n C_i \text{ is open} \implies \bigcup_{i=1}^n C_i \text{ is closed}

Since its complement is open, the finite union of closed sets is itself closed

Properties

De Morgan duality of the axioms

\text{Closed sets satisfy the DUAL axioms: } \emptyset, X \text{ closed}; \text{ arbitrary intersections of closed sets are closed}; \text{ finite unions of closed sets are closed.}

Condition: This follows directly from applying De Morgan's laws to the open-set axioms.

∅ and X are always clopen

\emptyset \text{ and } X \text{ are both open AND closed in every topological space } (X,\tau).

Closure as smallest closed superset

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

Condition: The closure always exists since X itself is a closed superset of A, and closed sets are closed under arbitrary intersection.

A is closed iff it contains all its limit points

A \text{ is closed} \iff \overline{A} = A \iff \text{every limit point of } A \text{ lies in } A.

Applications

Verification and safety properties in program analysis are often modeled as closed sets in a suitable topology (Scott/Alexandrov topologies), where 'closed' corresponds to properties preserved under limits of computation.

Worked Examples

Every point x>2 has a small interval (x-ε,x+ε) still inside A for small enough ε (e.g. ε=(x-2)/2), so A is open.
$A = (2, \infty) \text{ is open}$
Its complement (-∞,2] is closed (standard closed ray), but that doesn't make A itself closed — check A directly: A is not closed since it doesn't contain its limit point 2.
$2 \text{ is a limit point of } A \text{ not contained in } A$

Answer: A is open but not closed

Practice Problems

Difficulty 5/10

Which of the following sets is clopen (both open and closed) in every topological space (X, τ)?

Every singleton {x}∅ and XEvery finite setEvery countable set

Difficulty 6/10

Prove that the closed interval [a,b] is closed in the standard topology on ℝ, by showing its complement is open.

Common Mistakes

Common Mistake

Assuming 'closed' means 'not open' — that every set is one or the other.

Open and closed are independent properties, not opposites. [0,1) is neither open nor closed, while ∅ and X are always both (clopen) in any topological space.

Common Mistake

Thinking a finite union of closed sets could fail to be closed, by wrongly generalizing from the fact that infinite unions of closed sets CAN fail to be closed.

Finite unions of closed sets are always closed (dual to the finite-intersection axiom for open sets via De Morgan's law). It is infinite unions of closed sets that can fail — e.g. ∪ₙ[1/n, 1] = (0,1] is not closed, even though each [1/n,1] is closed.

Quiz

A set is CLOSED (in a topological space) exactly when:

it has finitely many pointsits complement is openit contains no boundaryit is bounded

Which statement about open and closed sets is TRUE?

Every set is either open or closedA set can be both open and closed, or neither (e.g. the half-open interval [0,1) is neither)Open and closed are oppositesOnly intervals can be open

Summary

In a topological space (X,τ), a set is open iff it belongs to τ, and closed iff its complement belongs to τ.
Open and closed are not opposites: a set may be open, closed, both (clopen — always true of ∅ and X), or neither (e.g. [0,1) in ℝ).
By De Morgan duality, closed sets obey the mirror-image axioms: ∅,X closed; arbitrary intersections of closed sets are closed; finite unions of closed sets are closed.
A set is closed exactly when it contains all its limit points, equivalently when it equals its own closure.
The closure of A is the smallest closed set containing A; the interior of A is the largest open set contained in A.

References

WebsiteWikipedia — Open set
BookMunkres, J. Topology, 2nd ed. Ch. 2.