Closed Set

Let $(X, \tau)$ be a topological space.

Definition. A subset $C \subseteq X$ is a closed set if its complement $X \setminus C$ is an open set, i.e., $X \setminus C \in \tau$.

Proposition. The collection of closed subsets of $X$ is closed under arbitrary intersections and finite unions. Moreover, $\emptyset$ and $X$ are both closed.

Proof sketch. By De Morgan’s laws: the complement of an arbitrary intersection of closed sets is an arbitrary union of open sets (hence open), and the complement of a finite union of closed sets is a finite intersection of open sets (hence open). $\square$

Definition. The closure of a subset $A \subseteq X$ is $\mathrm{cl}(A) = \bigcap \{ C \subseteq X : A \subseteq C \text{ and } C \text{ is closed} \}$, the smallest closed set containing $A$.

Proposition (Kuratowski axioms). The closure operator $\mathrm{cl} : \mathcal{P}(X) \to \mathcal{P}(X)$ satisfies:

1. $\mathrm{cl}(\emptyset) = \emptyset$,
2. $A \subseteq \mathrm{cl}(A)$ for all $A \subseteq X$,
3. $\mathrm{cl}(\mathrm{cl}(A)) = \mathrm{cl}(A)$ (idempotence),
4. $\mathrm{cl}(A \cup B) = \mathrm{cl}(A) \cup \mathrm{cl}(B)$.

Conversely, any operator $\mathrm{cl} : \mathcal{P}(X) \to \mathcal{P}(X)$ satisfying (1)–(4) determines a unique topology on $X$ whose closed sets are exactly the fixed points of $\mathrm{cl}$.

Proposition. A subset $C \subseteq X$ is closed if and only if it contains all its limit points: for every $x \in X$, if every open set containing $x$ meets $C$, then $x \in C$.

Proof sketch. $x \in \mathrm{cl}(A)$ iff every open neighbourhood of $x$ intersects $A$. So $C = \mathrm{cl}(C)$ iff $C$ contains all such points. $\square$

Examples.

1. $\mathbb{R}$ with the standard topology. Closed intervals $[a, b]$ are closed. The set $\mathbb{Z}$ is closed (its complement is a union of open intervals).
2. Discrete topology. Every subset is both open and closed.
3. Indiscrete topology. The only closed sets are $\emptyset$ and $X$.