# Kuratowski closure axioms

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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 ${\displaystyle X}$ be a set and ${\displaystyle {\mathcal {P}}(X)}$ its power set.
A Kuratowski Closure Operator is an assignment ${\displaystyle \operatorname {cl} :{\mathcal {P}}(X)\to {\mathcal {P}}(X)}$ with the following properties:[2]

1. ${\displaystyle \operatorname {cl} (\varnothing )=\varnothing }$ (Preservation of Nullary Union)
2. ${\displaystyle A\subseteq \operatorname {cl} (A){\text{ for every subset }}A\subseteq X}$ (Extensivity)
3. ${\displaystyle \operatorname {cl} (A\cup B)=\operatorname {cl} (A)\cup \operatorname {cl} (B){\text{ for any subsets }}A,B\subseteq X}$ (Preservation of Binary Union)
4. ${\displaystyle \operatorname {cl} (\operatorname {cl} (A))=\operatorname {cl} (A){\text{ for every subset }}A\subseteq X}$ (Idempotence)

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

A consequence of the third axiom is: ${\displaystyle A\subseteq B\Rightarrow \operatorname {cl} (A)\subseteq \operatorname {cl} (B)}$ (Preservation of Inclusion).[3]

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

${\displaystyle A\cup \operatorname {cl} (A)\cup \operatorname {cl} (\operatorname {cl} (B))=\operatorname {cl} (A\cup B)\setminus \operatorname {cl} (\varnothing ){\text{ for all subsets }}A,B\subseteq X.}$

## Connection to other axiomatizations of topology

### Induction of Topology

Construction
A closure operator naturally induces a topology as follows:
A subset ${\displaystyle C\subseteq X}$ is called closed if and only if ${\displaystyle \operatorname {cl} (C)=C}$.

Empty Set and Entire Space are closed:
By extensitivity, ${\displaystyle X\subseteq \operatorname {cl} (X)}$ and since closure maps the power set of ${\displaystyle X}$ into itself (that is, the image of any subset is a subset of ${\displaystyle X}$), ${\displaystyle \operatorname {cl} (X)\subseteq X}$ we have ${\displaystyle X=\operatorname {cl} (X)}$. Thus ${\displaystyle X}$ is closed.
The preservation of nullary unions states that ${\displaystyle \operatorname {cl} (\varnothing )=\varnothing }$. Thus ${\displaystyle \varnothing }$ is closed.

Arbitrary intersections of closed sets are closed:
Let ${\displaystyle {\mathcal {I}}}$ be an arbitrary set of indices and ${\displaystyle C_{i}}$ closed for every ${\displaystyle i\in {\mathcal {I}}}$.
By extensitivity, ${\displaystyle \bigcap _{i\in {\mathcal {I}}}C_{i}\subseteq \operatorname {cl} (\bigcap _{i\in {\mathcal {I}}}C_{i}).}$
Also, by preservation of inclusions, ${\displaystyle \bigcap _{i\in {\mathcal {I}}}C_{i}\subseteq C_{i}\forall i\in {\mathcal {I}}\Rightarrow \operatorname {cl} (\bigcap _{i\in {\mathcal {I}}}C_{i})\subseteq \operatorname {cl} (C_{i})=C_{i}\forall i\in {\mathcal {I}}\Rightarrow \operatorname {cl} (\bigcap _{i\in {\mathcal {I}}}C_{i})\subseteq \bigcap _{i\in {\mathcal {I}}}C_{i}.}$
Therefore, ${\displaystyle \bigcap _{i\in {\mathcal {I}}}C_{i}=\operatorname {cl} (\bigcap _{i\in {\mathcal {I}}}C_{i})}$. Thus ${\displaystyle \bigcap _{i\in {\mathcal {I}}}C_{i}}$ is closed.

Finite unions of closed sets are closed:
Let ${\displaystyle {\mathcal {I}}}$ be a finite set of indices and let ${\displaystyle C_{i}}$ be closed for every ${\displaystyle i\in {\mathcal {I}}}$.
From the preservation of binary unions and using induction we have ${\displaystyle \bigcup _{i\in {\mathcal {I}}}C_{i}=\operatorname {cl} (\bigcup _{i\in {\mathcal {I}}}C_{i})}$. Thus ${\displaystyle \bigcup _{i\in {\mathcal {I}}}C_{i}}$ 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: ${\displaystyle \operatorname {cl_{A}} (B)=A\cap \operatorname {cl_{X}} (B){\text{ for all }}B\subseteq A.}$[5]

### Recovering notions from topology

Closeness
A point ${\displaystyle p}$ is close to a subset ${\displaystyle A}$ iff ${\displaystyle p\in \operatorname {cl} (A)}$.

Continuity
A function ${\displaystyle f:X\to Y}$ is continuous at a point ${\displaystyle p}$ iff ${\displaystyle p\in \operatorname {cl} (A)\Rightarrow f(p)\in \operatorname {cl} (f(A))}$.