Definition 7

In a topological space $X$, the complements of the open sets of $X$ are called closed sets.

Taking complements, we find that the axioms $(O_I)$ and $(O_{II})$ take the following form:

$(O'_I)$ Every intersection of closed sets is a closed set.

$(O'_{II})$ Every finite union of closed sets is a closed set.

The empty set and the whole space $X$ are closed (and therefore both open and closed; cf. § 11).

On the rational line, every interval of the form $[a, \rightarrow[$ is a closed set, for its complement ]\leftarrow, a[ is open; likewise, every interval of the form ]\leftarrow, a] is a closed set; hence so is every bounded closed interval $[a, b]$, since it is the intersection of the intervals $[a, \rightarrow[$ and ]\leftarrow, b].

The set $Z$ of rational integers is closed in the rational line, since its complement $\bigcup_{n \in Z} ]n, n+1[$ is open.

A covering $(F_i)_{i \in I}$ of a subset $A$ of a topological space $X$ is said to be closed if each of the $F_i$ is closed in $X$.

A homeomorphism $f$ of a topological space $X$ onto a topological space $X'$ (no. 1) can be characterized as a bijection of $X$ onto $X'$ such that the image under $f$ of every closed subset of $X$ is a closed subset of $X'$ and the inverse image under $f$ of every closed subset of $X'$ is a closed subset of $X$.