Topological Closure#

Definition: Adherent Point

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

We say that \(p \in X\) is an adherent point / closure point of \(S\) if every neighborhood of \(p\) intersects \(S\).

\[\forall N(p): N(p) \cap S \ne \varnothing\]

Definition: Closure

The (topological) closure of \(S\) is the set of all its adherent points.

Notation

\[\overline{S}\]

Theorem: Closure via Interior and Boundary

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

The closure of \(S\) is the union of \(S\)'s iinterior and boundary:

\[\overline{S} = \operatorname{int} S \cup \partial S\]

Proof

TODO

Theorem: Closure is Closed

The closure of a subset \(S \subseteq X\) of a topological space \(X\) is always closed.

Proof

TODO

Theorem: Closure is a Superset

Every subset \(S \subseteq X\) of a topological space \(X\) is contained in its own closure:

\[S \subseteq \overline{S}\]

Proof

TODO

Theorem: Idempotence of Closure

The closure of the closure a subset \(S \subseteq X\) of a topological space \(X\) is always still the closure of \(S\):

\[\overline{\overline{S}} = \overline{S}\]

Proof

TODO

Theorem: Closure of Union

\[\overline{S_1\cup \cdots \cup S_n} = \overline{S_1} \cup \cdots \cup \overline{S_n}\]

Proof

TODO