Definition:Closed Set

Definition

Let $T = \left({X, \vartheta}\right)$ be a topological space.

Let $H \subseteq X$.

Then $H$ is closed (in $T \ $) iff its complement $X \setminus H$ is open in $T$.

That is, $H$ is closed iff $\left (X \setminus H \right ) \in \vartheta$.

That is, iff $X \setminus H$ is an element of the topology of $T$.

Relatively Closed

Let $T = \left({X, \vartheta}\right)$ be a topological space.

Let $A \subseteq B \subseteq X$.

Then $A$ is relatively closed in $B$ iff $A$ is closed in the relative topology of $B$.

Equivalently, $A$ is relatively closed in $B$ iff there is a closed set $C \subseteq X$ with $C \cap B = A$.

This is proved in Relatively Closed by Intersection with Closed Set.

Closed Point

The concept of a closed set can be sharpened to apply to individual points, as follows:

Let $a \in X$.

Then $a$ is closed (in $T \ $) iff $\left\{{a}\right\}$ is closed (in $T \ $).

Also see

• Closed Set contains All its Limit Points

• Results about Closed Sets can be found here.

Internationalization

Closed (in this context) is translated:

In French:

fermé

(literally: closed)

Sources

• Steven A. Gaal: Point Set Topology (1964)... (previous)... (next): $\S 1.1$: Definition $3$
• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$