# Kuratowski closure axioms

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

A topological space $(X,\operatorname{cl})$ is a set $X$ with a function

$\operatorname{cl}:\mathcal{P}(X) \to \mathcal{P}(X)$

called the closure operator where $\mathcal{P}(X)$ is the power set of $X$.

The closure operator has to satisfy the following properties for all $A, B\in\mathcal{P}(X)$

1. $A \subseteq \operatorname{cl}(A) \!$ (Extensivity)
2. $\operatorname{cl}(\operatorname{cl}(A)) = \operatorname{cl}(A) \!$ (Idempotence)
3. $\operatorname{cl}(A \cup B) = \operatorname{cl}(A) \cup \operatorname{cl}(B) \!$ (Preservation of binary unions)
4. $\operatorname{cl}(\varnothing) = \varnothing \!$ (Preservation of nullary unions)

If the second axiom, that of idempotence, is relaxed, then the axioms define a preclosure operator.

## Connection to other axiomatizations of topology

### Induction of a topology

A point $p$ is called close to $A$ in $(X,\operatorname{cl})$ if $p\in \operatorname{cl}(A)$

By defining a closure operator on $\mathcal{P}()$ a topology as defined usually (a set containing all open sets) is naturally induced as follows. A set $O \subset X$ is called open if and only if $\operatorname{cl}(X \setminus O) = X \setminus O$ and we construct $\tau := \{O | O \; \text{open} \}$. The pair $(X,\tau)$ then obeys the Open Sets Definition of a topological space:

Empty and complete sets are open: $\emptyset, X \in \tau$

By Extensitivity $X \subset \operatorname{cl}(X)$ and since $\operatorname{cl} \rightarrow \mathcal{P}(X)$ we know that $\operatorname{cl}(X) \subset X$, therefore $\operatorname{cl}(X) = X \Rightarrow \operatorname{cl}(X \setminus \emptyset) = X \setminus \emptyset \Leftrightarrow \emptyset \in \tau$. From Preservation of nullary unions follows similarly $X \in \tau$.

Any union of open sets is open:

Let $\mathcal{I}$ be a set of Indices and we unite every $A_i$ where $A_i$ is open for every $i \in \mathcal{I}$. By De Morgan's laws is

$A := \bigcup\limits_{i \in \mathcal{I}} A_i = X \setminus \bigcap\limits_{i \in \mathcal{I}} X \setminus A_i$ therefore
$X \setminus A = \bigcap\limits_{i \in \mathcal{I}} X \setminus A_i$.
$\Rightarrow X \setminus A \subset X \setminus A_i \forall i \in \mathcal{I}$
$\Rightarrow X \setminus A \cup X \setminus A_i = X \setminus A_i$

And by Preservation of binary unions:

$\Rightarrow \operatorname{cl} \left(X \setminus A \cup X \setminus A_i \right) = \operatorname{cl}(X \setminus A) \cup \operatorname{cl}(X \setminus A_i) = \operatorname{cl}(X \setminus A_i)$
$\Rightarrow \operatorname{cl}(X \setminus A) \subset \operatorname{cl}(X \setminus A_i) \forall i \in \mathcal{I}$
$\Rightarrow \operatorname{cl}(X\setminus A) \subset \bigcap\limits_{i \in \mathcal{I}} \operatorname{cl}(X \setminus A_i)= \bigcap\limits_{i \in \mathcal{I}} X \setminus A_i =X \setminus A$.

Hence $\operatorname{cl}(X\setminus A)\subset\ X\setminus A.$ And together with Extensivity follows $X\setminus A=\operatorname{cl}(X\setminus A)$ Therefore, A is open.

The intersection of any finite number of open sets is open.

Let $\mathcal{I}$ be a finite set of Indices with $A_i$ open $\forall i \in \mathcal{I}$.

$\bigcap\limits_{i \in \mathcal{I}} A_i = X \setminus \bigcup\limits_{i \in \mathcal{I}} X \setminus A_i=X \setminus \bigcup\limits_{i \in \mathcal{I}} \operatorname{cl}(X \setminus A_i)$

From the Preservation of nullary unions follows by induction:

$= X \setminus\operatorname{cl}\left( \bigcup\limits_{i \in \mathcal{I}} X \setminus A_i \right)$
$\Rightarrow X \setminus \bigcap\limits_{i \in \mathcal{I}} A_i=\operatorname{cl} \left( \bigcup\limits_{i \in \mathcal{I}} X \setminus A_i \right)$
$\Rightarrow X \setminus \bigcap\limits_{i \in \mathcal{I}} A_i$ is open.

### Recovering topological definitions

A function between two topological spaces

$f:(X,\operatorname{cl}) \to (X',\operatorname{cl}')$

is called continuous if for all subsets $A$ of $X$

$f(\operatorname{cl}(A)) \subset \operatorname{cl}'(f(A))$