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]

$\operatorname {cl}(\varnothing )=\varnothing$ (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 $p\in \operatorname {cl}(A)\Rightarrow f(p)\in \operatorname {cl}(f(A))$ .

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 T₁ 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, ISBN 0-12-429201-1
Pervin, William J. (1964), Foundations of General Topology, Academic Press

External links

Alternative Characterizations of Topological Spaces

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