# 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.^{[1]}

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

## Contents

## Definition[edit]

Let be a set and its power set.

A **Kuratowski Closure Operator** is an assignment with the following properties:^{[2]}

- (Preservation of Nullary Union)
- (Extensivity)
- (Preservation of Binary Union)
- (Idempotence)

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

A consequence of the third axiom is: (Preservation of Inclusion).^{[3]}

The four Kuratowski closure axioms can be replaced by a single condition, namely,^{[4]}

## Connection to other axiomatizations of topology[edit]

### Induction of Topology[edit]

**Construction**

A closure operator naturally induces a topology as follows:

A subset is called **closed** if and only if .

**Empty Set and Entire Space are closed:**

By extensitivity, and since closure maps the power set of into itself (that is, the image of any subset is a subset of ), we have . Thus is closed.

The preservation of nullary unions states that . Thus is closed.

**Arbitrary intersections of closed sets are closed:**

Let be an arbitrary set of indices and closed for every .

By extensitivity,

Also, by preservation of inclusions,

Therefore, . Thus is closed.

**Finite unions of closed sets are closed:**

Let be a finite set of indices and let be closed for every .

From the preservation of binary unions and using induction we have . Thus is closed.

### Induction of closure[edit]

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*: ^{[5]}

### Recovering notions from topology[edit]

**Closeness**

A point is close to a subset iff .

**Continuity**

A function is continuous at a point iff .

## Notes[edit]

**^**Kuratowski 1966, p. 38**^**Kuratowski (1966) has a fifth (optional) axiom stating that singleton sets are their own closures. He refers to topological spaces which satisfy all five axioms as T_{1}spaces in contrast to the more general spaces which only satisfy the four listed axioms.**^**Pervin 1964, p. 43**^**Pervin 1964, p. 42**^**Pervin 1964, p. 49, Theorem 3.4.3

## References[edit]

- Kuratowski, K. (1966) [1958],
*Topology Volume I*, Academic Press, ISBN 0-12-429201-1 - Pervin, William J. (1964),
*Foundations of General Topology*, Academic Press