Closure

Theorem: Closure

Let $(X,\tau )$ be a topological space let $A$ be a subset of $X$.

The union of the interior $\mathrm{int}A$ of $A$ with the boundary $\partial A$ of $A$ is closed.

PROOF

TODO

Definition: Closure

Let $(X,\tau )$ be a topological space and let $A$ be a subset of $X$.

The closure of $A$ is the union of its interior and its boundary.

$$\mathrm{int}A\cup \partial A$$

NOTATION

$$\overline{A}$$

Properties

Theorem: Closure is a Superset

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

Every subset $S$ is a subset of its own closure.

$$S\subseteq \overline{S}$$

PROOF

TODO

Theorem: Closure of a Closure

Let $(X,\tau )$ be a topological space and let $S$ be a subset of $X$.

The closure of the closure of $S$ is still the closure of $S$.

$$\overline{\overline{S}}=\overline{S}$$

PROOF

TODO

Theorem: Closure of a Union

Let $(X,\tau )$ be a topological space and let $A$ and $B$ be two arbitrary subsets of $X$.

The closure of the union of $A$ and $B$ is the union of the closures of $A$ and $B$.

$$\overline{A\cup B}=\overline{A}\cup \overline{B}$$

PROOF

TODO

Theorem: Closure of the Empty Set

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

The closure of the empty set is itself.

$$\overline{\varnothing }=\varnothing$$

PROOF

TODO