closure space

Call a set $X$ with a closure operator defined on it a closure space.

Every topological space is a closure space, if we define the closure operator of the space as a function that takes any subset to its closure. The converse is also true:

Proposition 1.

Let $X$ be a closure space with $c$ the associated closure operator. Define a “closed set” of $X$ as a subset $A$ of $X$ such that $A^{c}=A$ , and an “open set” of $X$ as the complement of some closed set of $X$ . Then the collection $\mathcal{T}$ of all open sets of $X$ is a topology on $X$ .

Proof.

Since $\varnothing^{c}=\varnothing$ , $\varnothing$ is closed. Also, $X\subseteq X^{c}$ and $X^{c}\subseteq X$ imply that $X^{c}=X$ , or $X$ is closed. If $A,B\subseteq X$ are closed, then $(A\cup B)^{c}=A^{c}\cup B^{c}=A\cup B$ is closed as well. Finally, suppose $A_{i}$ are closed. Let $B=\bigcap A_{i}$ . For each $i$ , $A_{i}=B\cup A_{i}$ , so $A_{i}=A_{i}^{c}=(B\cup A_{i})^{c}=B^{c}\cup A_{i}^{c}=B^{c}\cup A_{i}$ . This means $B^{c}\subseteq A_{i}$ , or $B^{c}\subseteq\bigcap A_{i}=B$ . But $B\subseteq B^{c}$ by definition, so $B=B^{c}$ , or that $\bigcap A_{i}$ is closed. ∎

$\mathcal{T}$ so defined is called the closure topology of $X$ with respect to the closure operator $c$ .

Remarks.

1.

A closure space can be more generally defined as a set $X$ together with an operator $\operatorname{cl}:P(X)\to P(X)$ such that $\operatorname{cl}$ satisfies all of the Kuratowski’s closure axioms where the equal sign “ $=$ ” is replaced with set inclusion “ $\subseteq$ ”, and the preservation of $\varnothing$ is no longer assumed.
2.
Even more generally, a closure space can be defined as a set $X$ and an operator $\operatorname{cl}$ on $P(X)$ such that
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  $A\subseteq\operatorname{cl}(A)$ ,
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  $\operatorname{cl}(\operatorname{cl}(A))\subseteq\operatorname{cl}(A)$ , and
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  $\operatorname{cl}$ is order-preserving, i.e., if $A\subseteq B$ , then $\operatorname{cl}(A)\subseteq\operatorname{cl}(B)$ .
It can be easily deduced that $\operatorname{cl}(A)\cup\operatorname{cl}(B)\subseteq\operatorname{cl}(A\cup B)$ . In general however, the equality fails. The three axioms above can be shown to be equivalent to a single axiom:

$A\subseteq\operatorname{cl}(B)\quad\mbox{ iff }\quad\operatorname{cl}(A)% \subseteq\operatorname{cl}(B).$
3.

In a closure space $X$ , a subset $A$ of $X$ is said to be closed if $\operatorname{cl}(A)=A$ . Let $C(X)$ be the set of all closed sets of $X$ . It is not hard to see that if $C(X)$ is closed under $\cup$ , then $\operatorname{cl}$ “distributes over” $\cup$ , that is, we have the equality $\operatorname{cl}(A)\cup\operatorname{cl}(B)=\operatorname{cl}(A\cup B)$ .
4.

Also, $\operatorname{cl}(\varnothing)$ is the smallest closed set in $X$ ; it is the bottom element in $C(X)$ . This means that if there are two disjoint closed sets in $X$ , then $\operatorname{cl}(\varnothing)=\varnothing$ . This is equivalent to saying that $\varnothing$ is closed whenever there exist $A,B\subseteq X$ such that $\operatorname{cl}(A)\cap\operatorname{cl}(B)=\varnothing$ .
5.

Since the distributivity of $\operatorname{cl}$ over $\cup$ does not hold in general, and there is no guarantee that $\operatorname{cl}(\varnothing)=\varnothing$ , a closure space under these generalized versions is a more general system than a topological space.

References

1 N. M. Martin, S. Pollard: Closure Spaces and Logic, Springer, (1996).

Title	closure space
Canonical name	ClosureSpace
Date of creation	2013-03-22 16:48:08
Last modified on	2013-03-22 16:48:08
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	12
Author	CWoo (3771)
Entry type	Derivation
Classification	msc 54A05
Defines	closure topology