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In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms which 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, in a slightly different form that applied only to Hausdorff spaces.

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 $𝒫(X)$ its power set.
A Kuratowski Closure Operator is an assignment $\mathrm{cl}:𝒫(X)\to 𝒫(X)$ with the following properties:

1. $\mathrm{cl}(\varnothing )=\varnothing$ (Preservation of Nullary Union)
2. $A\subseteq \mathrm{cl}(A)$ (Extensivity)
3. $\mathrm{cl}(A\cup B)=\mathrm{cl}(A)\cup \mathrm{cl}(B)$ (Preservation of Binary Union)
4. $\mathrm{cl}(\mathrm{cl}(A))=\mathrm{cl}(A)$ (Idempotence)

If the last axiom, Idempotence, is omitted, then the axioms define a Preclosure Operator.
A consequence from the third axiom is: $A\subseteq B\Rightarrow \mathrm{cl}(A)\subseteq \mathrm{cl}(B)$ (Preservation of Inclusion)

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 $\mathrm{cl}(C)=C$.

Empty Set and Entire Space are closed:
By Extensitivity $X\subseteq \mathrm{cl}(X)$ and since Closure maps into itself $\mathrm{cl}(X)\subseteq X$ we have $X=\mathrm{cl}(X)$. Thus $X$ is closed.
By Preservation of Nullary Unions follows $\mathrm{cl}(\varnothing )=\varnothing$. Thus $\varnothing$ is closed

Arbitrary intersections of closed sets is closed:
Let $ℐ$ be an arbitrary set of indices and $C_i$ closed for every $i\in ℐ$.
Then by Extensitivity: $\bigcap _{i\in ℐ}{C}_i\subseteq \mathrm{cl}(\bigcap _{i\in ℐ}{C}_i)$
Also by Preservation of Inclusions: $\bigcap _{i\in ℐ}{C}_i\subseteq C_i\mathrm{\forall }i\in ℐ\Rightarrow \mathrm{cl}(\bigcap _{i\in ℐ}{C}_i)\subseteq \mathrm{cl}(C_i)=C_i\mathrm{\forall }i\in ℐ\Rightarrow \mathrm{cl}(\bigcap _{i\in ℐ}{C}_i)\subseteq \bigcap _{i\in ℐ}{C}_i$
And therefore $\bigcap _{i\in ℐ}{C}_i=\mathrm{cl}(\bigcap _{i\in ℐ}{C}_i)$. Thus $\bigcap _{i\in ℐ}{C}_i$ is closed.

Finite unions of closed sets is closed:
Let $ℐ$ be a finite set of indices and $C_i$ closed for every $i\in ℐ$.
From the Preservation of binary unions and by induction we have $\bigcup _{i\in ℐ}{C}_i=\mathrm{cl}(\bigcup _{i\in ℐ}{C}_i)$. Thus $\bigcup _{i\in ℐ}{C}_i$ is closed.

Induction of Closure

The induced topology reinduces a closure which agrees with the original closure: $\overline{A}=\mathrm{cl}(A)$
For a proof see Alternative Characterizations of Topological Spaces.

Recovering Notions from Topology

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

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

See also

• Preclosure operator

External links

• Alternative Characterizations of Topological Spaces