Closed Sets and Closure

Closed sets are the complement of open sets, and the closure operation provides one of the most useful tools for studying the structure of topological spaces. The interplay between open sets, closed sets, and closure is at the heart of point-set topology.

Closed Sets

Definition1.5Closed Set

Let $(X, \tau)$ be a topological space. A subset $C \subseteq X$ is closed if its complement $X \setminus C$ is open, i.e., $X \setminus C \in \tau$ .

By De Morgan's laws applied to the topology axioms, closed sets satisfy the following dual properties:

$\emptyset$ and $X$ are closed.
Arbitrary intersections of closed sets are closed: if $\{C_\alpha\}_{\alpha \in A}$ is a family of closed sets, then $\bigcap_{\alpha \in A} C_\alpha$ is closed.
Finite unions of closed sets are closed: if $C_1, \ldots, C_n$ are closed, then $\bigcup_{i=1}^n C_i$ is closed.

ExampleClosed Sets in the Real Line

In $\mathbb{R}$ with the standard topology:

Closed intervals $[a, b]$ are closed sets.
Singleton sets $\{x\}$ are closed (their complement $(-\infty, x) \cup (x, \infty)$ is open).
The set $\mathbb{Z}$ of integers is closed.
The set $[0, 1)$ is neither open nor closed. Its complement $(-\infty, 0) \cup [1, \infty)$ is not open since $[1, \infty)$ fails to be open at $x = 1$ .

RemarkClopen Sets

A set that is both open and closed is called clopen. In any topological space, $\emptyset$ and $X$ are always clopen. A space $X$ is connected if and only if $\emptyset$ and $X$ are the only clopen subsets (see Chapter 4).

The Closure Operator

Definition1.6Closure

Let $(X, \tau)$ be a topological space and $A \subseteq X$ . The closure of $A$ , denoted $\overline{A}$ or $\operatorname{cl}(A)$ , is the smallest closed set containing $A$ : $\overline{A} = \bigcap \{C \subseteq X : C \text{ is closed and } A \subseteq C\}.$

Since arbitrary intersections of closed sets are closed, $\overline{A}$ is indeed a well-defined closed set.

Definition1.7Limit Point

Let $(X, \tau)$ be a topological space and $A \subseteq X$ . A point $x \in X$ is a limit point (or accumulation point or cluster point) of $A$ if every open set $U$ containing $x$ satisfies: $(U \setminus \{x\}) \cap A \neq \emptyset.$

The set of all limit points of $A$ is called the derived set of $A$ , denoted $A'$ or $A^d$ .

The closure can be characterized in terms of limit points:

$\overline{A} = A \cup A'.$

ExampleClosure Computations

In $\mathbb{R}$ with the standard topology:

$\overline{(a, b)} = [a, b]$ for $a < b$ .
$\overline{\mathbb{Q}} = \mathbb{R}$ (the rationals are dense in $\mathbb{R}$ ).
$\overline{\{1/n : n \in \mathbb{Z}^+\}} = \{1/n : n \in \mathbb{Z}^+\} \cup \{0\}$ , since $0$ is the only limit point.
In $\mathbb{R}$ with the cofinite topology, $\overline{A} = X$ for every infinite set $A$ , and $\overline{A} = A$ for every finite set $A$ .

Kuratowski Closure Axioms

Definition1.8Kuratowski Closure Axioms

The closure operator $\operatorname{cl}: \mathcal{P}(X) \to \mathcal{P}(X)$ satisfies the following Kuratowski closure axioms:

$\operatorname{cl}(\emptyset) = \emptyset$ (preservation of the empty set).
$A \subseteq \operatorname{cl}(A)$ (extensiveness).
$\operatorname{cl}(\operatorname{cl}(A)) = \operatorname{cl}(A)$ (idempotence).
$\operatorname{cl}(A \cup B) = \operatorname{cl}(A) \cup \operatorname{cl}(B)$ (preservation of binary unions).

Conversely, any operator $\operatorname{cl}: \mathcal{P}(X) \to \mathcal{P}(X)$ satisfying these four axioms determines a unique topology on $X$ whose closed sets are exactly the fixed points of $\operatorname{cl}$ .

Boundary and Dense Sets

Definition1.9Boundary

The boundary (or frontier) of a set $A$ in a topological space $(X, \tau)$ is: $\partial A = \overline{A} \cap \overline{X \setminus A} = \overline{A} \setminus \operatorname{int}(A).$

A point $x \in \partial A$ if and only if every open set containing $x$ intersects both $A$ and $X \setminus A$ .

Definition1.10Dense Set

A subset $A$ of a topological space $(X, \tau)$ is dense in $X$ if $\overline{A} = X$ . Equivalently, $A$ is dense if and only if every nonempty open set $U \in \tau$ satisfies $U \cap A \neq \emptyset$ .

A topological space is separable if it admits a countable dense subset.

ExampleDense Sets

$\mathbb{Q}$ is dense in $\mathbb{R}$ (standard topology). Thus $\mathbb{R}$ is separable.
$\mathbb{R} \setminus \mathbb{Q}$ (the irrationals) is also dense in $\mathbb{R}$ .
In the cofinite topology on an uncountable set $X$ , every infinite subset is dense.
In the discrete topology, the only dense subset is $X$ itself.

RemarkNowhere Dense Sets

A set $A$ is nowhere dense if $\operatorname{int}(\overline{A}) = \emptyset$ , equivalently if $\overline{A}$ has empty interior. The Baire category theorem (for complete metric spaces or locally compact Hausdorff spaces) states that a countable union of nowhere dense sets has empty interior. This provides a powerful tool for existence proofs in analysis.