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.

Contents

Definition

Notes

Recovering topological definitions

See also

Definition

A topological space $(X,operatorname\{cl\})$ is a set $X$ with a function
:$operatorname\{cl\}:mathcal\{P\}(X)\; o\; mathcal\{P\}(X)$
called the 'closure operator' where $mathcal\{P\}(X)$ is the power set of $X$.
The closure operator has to satisfy the following properties
# $A\; subseteq\; operatorname\{cl\}(A)\; !$ (Extensivity)
# $operatorname\{cl\}(operatorname\{cl\}(A))\; =\; operatorname\{cl\}(A)\; !$ (Idempotence)
# $operatorname\{cl\}(A\; cup\; B)\; =\; operatorname\{cl\}(A)\; cup\; operatorname\{cl\}(B)\; !$ (Preservation of binary unions)
# (Preservation of nullary unions)
If the second axiom, that of idempotence, is relaxed, then the axioms define a praclosure.

Notes

Axioms (3) and (4) can be generalised (using a proof by mathematical induction) to the equivalent single statement:
:$operatorname\{cl\}(A\_\{1\}\; cup\; cdots\; cup\; A\_\{n\})\; =\; operatorname\{cl\}(A\_\{1\})\; cup\; cdots\; cup\; operatorname\{cl\}(A\_\{n\}),\; n\; geq\; 0\; !$ (Preservation of finitary unions).

Recovering topological definitions

A function between two topological spaces
:$f:(X,operatorname\{cl\})\; o\; (X\text{'},operatorname\{cl\}\text{'})$
is called 'continuous' if for all subsets $A$ of $X$
:$f(operatorname\{cl\}(A))\; subset\; operatorname\{cl\}\text{'}(f(A))$
A point $p$ is called 'close' to $A$ in $(X,operatorname\{cl\})$ if $pin\; operatorname\{cl\}(A)$
$A$ is called 'closed' in $(X,operatorname\{cl\})$ if $A=operatorname\{cl\}(A)$. In other words the closed sets of $X$ are the fixed points of the closure operator.

See also

★ Praclosure operator