Zentrum

In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms which can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition. They were first introduced by Kazimierz Kuratowski, in a slightly different form that applied only to Hausdorff spaces.

A similar set of axioms can be used to define a topological structure using only the dual notion of interior operator.

Definition

Let $X$ be a set and $𝒫 (X)$ its power set.
A Kuratowski Closure Operator is an assignment $cl : 𝒫 (X) \to 𝒫 (X)$ with the following properties:

If the last axiom, Idempotence, is omitted, then the axioms define a Preclosure Operator.
A consequence from the third axiom is: $A \subseteq B \Rightarrow cl (A) \subseteq cl (B)$ (Preservation of Inclusion)

Construction
A closure operator naturally induces a topology as follows:
A subset $C \subseteq X$ is called closed if and only if $cl (C) = C$ .

Empty Set and Entire Space are closed:
By Extensitivity $X \subseteq cl (X)$ and since Closure maps into itself $cl (X) \subseteq X$ we have $X = cl (X)$ . Thus $X$ is closed.
By Preservation of Nullary Unions follows $cl (\emptyset) = \emptyset$ . Thus $\emptyset$ is closed

Finite unions of closed sets is closed:
Let $ℐ$ be a finite set of indices and $C_{i}$ closed for every $i \in ℐ$ .
From the Preservation of binary unions and by induction we have $⋃_{i \in ℐ} C_{i} = cl (⋃_{i \in ℐ} C_{i})$ . Thus $⋃_{i \in ℐ} C_{i}$ is closed.

The induced topology reinduces a closure which agrees with the original closure: $\bar{A} = cl (A)$
For a proof see Alternative Characterizations of Topological Spaces.

Closeness
A point $p$ is close to a subset $A$ iff $p \in cl (A)$ .

Continuity
A function $f : X \to Y$ is continuous at a point $p$ iff $p \in cl (A) \Rightarrow f (p) \in cl (f (A))$ .