Definition:Closure (Topology)

Definition

Let $T$ be a topological space, and let $H \subseteq T$.

Then the closure of $H$ is defined as:

$\operatorname{cl} \left({H}\right)$ is the union of $H$ and its limit points.

From the definition of derived set, this is equivalent to:

$\operatorname{cl} \left({H}\right) = H \cup H'$

where $H'$ is the derived set of $H$.

Notation

The closure of $H$ is variously denoted:

• $\operatorname{cl} \left({H}\right)$
• $\operatorname{Cl} \left({H}\right)$
• $\overline H$
• $H^-$

Of these, it can be argued that $\overline H$ has more ambiguity problems than the others, as it is also frequently used for the set complement.

$\operatorname{cl} \left({H}\right)$ and $\operatorname{Cl} \left({H}\right)$ are regarded by some as cumbersome, but they have the advantage of being clear.

$H^-$ is neat and compact, but has the disadvantage of being relatively obscure.

On this website, $\operatorname{cl} \left({H}\right)$ and $H^-$ are the notations of choice.

Equivalent Definitions

The following definitions for closure are equivalent to the above:

• $\displaystyle H^- := \bigcap_{H \subseteq K \subseteq T, K \text{ closed}} K$
• $H^-$ is the smallest closed set that contains $H$
• $H^-$ is the union of $H$ and its boundary
• $H^-$ is the union of all isolated points of $H$ and all limit points of $H$.

This fact is demonstrated in Equivalent Definitions for Topological Closure.

Also see

• Interior

• Results about set closures can be found here.

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$: Closures and Interiors