# Charecterization of Topologies via Galois Connections

Let $$X$$ and $$Y$$ be two sets and let $$F: \mathcal{P}(X) \to \mathcal{P}(Y)$$ and $$G: \mathcal{P}(Y) \to \mathcal{P}(X)$$ be two set functions that satisfy $$F(A) \subseteq B \iff A \subseteq G(B). \tag{1}$$

That is, the pair $$F$$ and $$G$$ constitute a Galois connection. It is well known that $$c = G\circ F$$ is a(n abstract) closure operator and $$i = F \circ G$$ is an (abstract) interior operator. I.e., $$c$$ (resp $$i$$) satisfies:

• $$A \subseteq c(A)$$ (resp., $$i(A) \subseteq A$$) for all $$A \subseteq X$$.
• $$c(c(A)) = c(A)$$ (resp., $$i(i(A)) = i(A)$$), for all $$A \subseteq X$$.
• $$A \subseteq B$$ implies $$c(A) \subseteq c(B)$$ (resp., $$i(A) \subseteq i(B)$$), for all $$A, B \subseteq X$$.

A topological closure operator also satisfies

• $$c(A \cup B) = c(A) \cup c(B)$$ for all $$A, B \subseteq X$$.

A topological interior operator also satisfies

• $$i(A \cap B) = i(A) \cap i(B)$$ for all $$A, B \subseteq X$$.

Question: are there intelligible conditions on $$F$$ and $$G$$ such that $$c$$ is a topological closure operator and $$i$$ and topological interior operator? Is there a characterization of topology via Galois connections?

A bit more motivation: if $$Z$$ is a topological space, and $$X$$ is the set of closed sets and $$Y$$ the set of open sets, then $$F: A \mapsto int(A)$$ (the interior of $$A$$) and $$G: V \mapsto cl(B)$$ (the closure), then $$F$$ and $$G$$ satisfy (1). Then $$i$$ sets $$A$$ to $$int(cl(A))$$, the regular interior of $$A$$. Now, the intersection of regular open sets is regular open, so it would seem (naively) that $$F \circ G$$ is a topological interior operator, despite regular opens sets not being closed under unions.

• Do you have some examples of nontopological closure / interior operators? – Jeroen van der Meer Jan 18 at 21:49
• I believe the map $X$ to the convex hull of $X$ is a non topological closure operator (over say, $\mathbb{R}$). – 201p Jan 19 at 8:55