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.

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

See also

• Preclosure operator

References

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External links

• Alternative Characterizations of Topological Spaces