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# Kuratowski closure axioms

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 Title: Kuratowski closure axioms Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Kuratowski closure axioms

In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms that 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.

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

## Contents

• Definition 1
• Connection to other axiomatizations of topology 2
• Induction of Topology 2.1
• Induction of closure 2.2
• Recovering notions from topology 2.3
• Notes 4
• References 5

## Definition

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

1. \operatorname{cl}(\varnothing) = \varnothing (Preservation of Nullary Union)
2. A \subseteq \operatorname{cl}(A) \text{ for every subset }A \subseteq X (Extensivity)
3. \operatorname{cl}(A \cup B) = \operatorname{cl}(A) \cup \operatorname{cl}(B) \text{ for any subsets }A,B \subseteq X (Preservation of Binary Union)
4. \operatorname{cl}(\operatorname{cl}(A)) = \operatorname{cl}(A) \text{ for every subset }A \subseteq X (Idempotence)

If the last axiom, idempotence, is omitted, then the axioms define a preclosure operator.

A consequence of the third axiom is: A \subseteq B \Rightarrow \operatorname{cl}(A) \subseteq \operatorname{cl}(B) (Preservation of Inclusion).

The four Kuratowski closure axioms can be replaced by a single condition, namely,

A \cup \operatorname{cl}(A) \cup \operatorname{cl}(\operatorname{cl}(B)) = \operatorname{cl}(A \cup B) \setminus \operatorname{cl}(\varnothing) \text{ for all subsets }A, B \subseteq X.

## Connection to other axiomatizations of topology

### Induction of Topology

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

Empty Set and Entire Space are closed:
By extensitivity, X\subseteq\operatorname{cl}(X) and since closure maps the power set of X into itself (that is, the image of any subset is a subset of X), \operatorname{cl}(X)\subseteq X we have X = \operatorname{cl}(X). Thus X is closed.
The preservation of nullary unions states that \operatorname{cl}(\varnothing) = \varnothing . Thus \varnothing is closed.

Arbitrary intersections of closed sets are closed:
Let \mathcal{I} be an arbitrary set of indices and C_i closed for every i\in\mathcal{I}.
By extensitivity, \bigcap_{i\in\mathcal{I}}C_i \subseteq \operatorname{cl}(\bigcap_{i\in\mathcal{I}}C_i).
Also, by preservation of inclusions, \bigcap_{i\in\mathcal{I}}C_i \subseteq C_i \forall i\in\mathcal{I} \Rightarrow \operatorname{cl}(\bigcap_{i\in\mathcal{I}}C_i) \subseteq \operatorname{cl}(C_i) = C_i \forall i\in\mathcal{I} \Rightarrow \operatorname{cl}(\bigcap_{i\in\mathcal{I}}C_i) \subseteq \bigcap_{i\in\mathcal{I}}C_i.
Therefore, \bigcap_{i\in\mathcal{I}}C_i = \operatorname{cl}(\bigcap_{i\in\mathcal{I}}C_i) . Thus \bigcap_{i\in\mathcal{I}}C_i is closed.

Finite unions of closed sets are closed:
Let \mathcal{I} be a finite set of indices and let C_i be closed for every i\in\mathcal{I} .
From the preservation of binary unions and using induction we have \bigcup_{i\in\mathcal{I}}C_i = \operatorname{cl}(\bigcup_{i\in\mathcal{I}}C_i) . Thus \bigcup_{i\in\mathcal{I}}C_i is closed.

### Induction of closure

In any induced topology (relative to the subset A) the closed sets induce a new closure operator that is just the original closure operator restricted to A: \operatorname{cl_A}(B) = A \cap \operatorname{cl_X}(B) \text{ for all } B \subseteq A. 

### Recovering notions from topology

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

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