Relationship Between Closure Algebras and Topological Spaces

Question

Is it possible to construct the elementary theory of topology by extending the elementary theory of closure algebra?

Answer 1

There is a quite old theory on so-called "closure spaces", i.e. a set with an operator $\operatorname{cl}: \mathscr{P}(X) \to \mathscr{P}(X)$ such that the following axioms are fulfilled:

1. $\operatorname{cl}(\emptyset)=\emptyset$.
2. $\forall A: A \subseteq \operatorname{cl}(A)$.
3. $\forall A,B: \operatorname{cl}(A \cup B)=\operatorname{cl}(A) \cup \operatorname{cl}(B)$.

And in a topological space we can define such a closure operation by defining

$$\operatorname{cl}(A)=\bigcap \{B : A \subseteq B \text{ and } B \text{ closed }\}$$

(among many equivalent ways).

One can develop a theory close to topology on closure spaces (continuity for $f: X \to Y$ between closure spaces is defined as $\forall A\subseteq X: f[\operatorname{cl}_X(A)] \subseteq \operatorname{cl}_Y(f[A])$, e.g.)

Not all closure spaces arise from topological spaces in the above way, but we can characterise those that do by the extra condition

$$\forall A: \operatorname{cl}(\operatorname{cl}(A)) = \operatorname{cl}(A)$$

and closure spaces that obey this are topological closure spaces and fully equivalent with topological spaces. But you can develop a lot of the theory in just closure spaces without that last condition, Cech wrote a standard reference book on it quite a long time ago. They're not as popular as a research area any more, is my impression.