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:

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 $\cl:P(X)\to P(X)$ such that $\cl$ 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 $\cl$ on $P(X)$ such that
   • $A\subseteq \cl(A)$ ,
   • $\cl(\cl(A))\subseteq \cl(A)$ , and
   • $\cl$ is order-preserving, i.e., if $A\subseteq B$ , then $\cl(A)\subseteq \cl(B)$ .
   It can be easily deduced that $\cl(A)\cup \cl(B)\subseteq \cl(A\cup B)$ . In general however, the equality fails. The three axioms above can be shown to be equivalent to a single axiom: $$A\subseteq \cl(B)\quad\mbox{ iff }\quad\cl(A)\subseteq \cl(B).$$
3. In a closure space $X$ , a subset $A$ of $X$ is said to be closed if $\cl(A)=A$ . 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 $\cl$ ``distributes over'' $\cup$ , that is, we have the equality $\cl(A)\cup \cl(B)= \cl(A\cup B)$ .
4. Also, $\cl(\varnothing)$ 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 $\cl(\varnothing)=\varnothing$ . This is equivalent to saying that $\varnothing$ is closed whenever there exist $A,B\subseteq X$ such that $\cl(A)\cap\cl(B)=\varnothing$ .
5. Since the distributivity of $\cl$ over $\cup$ does not hold in general, and there is no guarantee that $\cl(\varnothing)=\varnothing$ , a closure space under these generalized versions is a more general system than a topological space.

Bibliography

1

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