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 1

Connection to other axiomatizations of topology 2

Induction of Topology 2.1

Induction of closure 2.2

Recovering notions from topology 2.3

See also 3

Notes 4

References 5

External links 6
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:^{[2]}

\operatorname{cl}(\varnothing) = \varnothing (Preservation of Nullary Union)

A \subseteq \operatorname{cl}(A) \text{ for every subset }A \subseteq X (Extensivity)

\operatorname{cl}(A \cup B) = \operatorname{cl}(A) \cup \operatorname{cl}(B) \text{ for any subsets }A,B \subseteq X (Preservation of Binary Union)

\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).^{[3]}
The four Kuratowski closure axioms can be replaced by a single condition, namely,^{[4]}

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. ^{[5]}
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)).
See also
Notes

^ 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

Kuratowski, K. (1966) [1958], Topology Volume I, Academic Press,

Pervin, William J. (1964), Foundations of General Topology, Academic Press
External links

Alternative Characterizations of Topological Spaces
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