Let $(X,\leq)$ and $(Y,\leq)$ by partially ordered sets. Recall that a(n antitone) **Galois connection** between $X$ and $Y$ is a pair of order-reversing maps

$\Phi: X \rightarrow Y, \ \Psi: Y \rightarrow X$

such that for all $x \in X,\ y \in Y$, $x \leq \Psi(y) \iff \Phi(x) \geq y$.

In this situation, the composite $\Psi \circ \Phi$ (resp. $\Phi \circ \Psi$) is a **closure operator** $\operatorname{cl}$ on $X$ (resp. $Y$): that is,

(C1) For all $x \in X$, $x \leq \operatorname{cl}(x)$.

(C2) For all $x_1,x_2 \in X$, $x_1 \leq x_2 \implies \operatorname{cl}(x_1) \leq \operatorname{cl}(x_2)$.

(C3) For all $x \in X$, $\operatorname{cl}(\operatorname{cl}(x)) = \operatorname{cl}(x)$.

Let us consider the special case where $X$ is the power set of some set $\mathbb{X}$, partially ordered by inclusion. In order for the above closure operator to define a topology on $\mathbb{X}$ -- i.e., be a "Kuratowski closure operator" -- we need also

(C4) For all $x,y \in X$, $\operatorname{cl}(x \cup y) = \operatorname{cl}(x) \cup \operatorname{cl}(y)$.

It is easy to see that (C4) is not automatic for the closure operator associated to a Galois connection -- in fact, every closure operator is induced by some Galois connection, and there are lots of closure operators which do not satisfy (C4).

However, in practice, it seems quite often to be the case that at least one of the two closure operators induced by a Galois connection is a topological closure operator. Examples:

(1) The Krull topology on the automorphism group of an infinite Galois extension (coming from the usual Galois correspondence).

(2) For a field $K$, the Zariski topology on $K^n$ (coming from the Galois connection arising in the Nullstellensatz).

(3) For a field $K$, the Harrison topology on the real spectrum of $K$ (coming from the Galois connection induced by the "incidence relation" $x \in P$ for $x$ an element of $K$ and $P$ an ordering on $K$).

Question 1: Is there some natural abstract condition on the Galois connection one can impose in order to ensure that one of the two closure operators is a Kuratowski closure operator?

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Question 2: Are there are other examples of topologies arising from Galois connections in an interesting way? [As above,

everytopology arises from some Galois connection, yet in a tautological way.]