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.

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]

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).^[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

• Preclosure operator

Notes

1. Kuratowski 1966, p. 38
2. 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 T₁ spaces in contrast to the more general spaces which only satisfy the four listed axioms.
3. Pervin 1964, p. 43
4. Pervin 1964, p. 42
5. Pervin 1964, p. 49, Theorem 3.4.3

References

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

External links

• Alternative Characterizations of Topological Spaces