Closure (Topology)

up::Topological Space

Given a topological space $(X,\tau )$ and a set $Z\subset X$, we denote its interior as closure as the intersection of all closed sets contained inside it.

It’s essentially “crunching $Z$ from the outside-in” with closed sets.

$$\overline{Z}:=\bigcap _{\begin{array}{c}F\in \mathcal{F}(\tau )\\ Z\subseteq F\end{array}}F$$

Properties

• Closure preserves subset ordering
• All sets are contained inside their closure
• The closure of a finite union is the union of the closures
• The closure of an arbitrary intersection is contained in the intersection of the closures

Closures from Interiors

Note that, given an “interior operator”, one can infer its closure, since The closure is the complement of the interior of the complement.

Relation to continuity

Via the definition of a Topologically Continuous Function $f:X\to Y$, one knows that it “preimages” open sets to open sets, and also closed sets to closed sets.

Denote the closure of a set $A$ in $X$‘s topology as as $C{l}_X(A)$. Then, phrased another way, Continuous functions map close inputs to close outputs.

---

References

• Notas para um Curso de Física-Matemática, João Carlos Alves Barata. Cap. 27