Axiomatic definition of Katětov closure operator

Question

In the book "Categorical Structure of Closure Operators with Applications to Topology" by Dikranjan and Tholen a Katětov closure operator is defined in terms of filter covergence:

$k_X(M):=\{x \in X: \exists \mathcal{F} \; \mathcal{F} \to x \land M \in \mathcal{F}\}$,

is there an axiomatic definition of closures that correspond to the convergence spaces similar to that of axioms for Kuratowski (for topological spaces) and Čech (for pretopological spaces) closures?

Answer 1

There is a discussion of the relation between convergence spaces and closeness spaces in the paper

S. Kasahara, Closeness spaces and convergence spaces, Proc. Japan Acad., 50(4) (1974), 303-308.

The structure of a closeness spaces is determined by a family of semiclosure operators - which are weaker than Cech closure operators. On the other hand, if a closure space is determined by family consisting of a single semiclosure operator, then this must be a Cech closure operator.

Answer 2

This is a partial answer. Let $X$ be a compact Hausdorff space and let $\mathcal{U}$ be an ultrafilter on a set $I$. Then any function $\phi: I \to X$ induces an ultrafilter on $X$, namely $\{A \subseteq X: \phi^{-1}(A) \in \mathcal{U}\}$. In a compact Hausdorff space every ultrafilter converges to exactly one point. So for each $I$ and $\mathcal{U}$ we get an infinitary "operation" $f_{I,\mathcal{U}}$ which takes an $I$-tuple of elements of $X$ (i.e., a function from $I$ into $X$) to a single element of $X$.

Compact Hausdorff spaces can be axiomatized in terms of the existence of these operations $f_{I,\mathcal{U}}$ by the three axioms

1. (triviality) if $I = \{i\}$ consists of a single point then $f_{I,\mathcal{U}}(\phi) = \phi(i)$

2. (restriction) if $A \in \mathcal{U}$ then $f_{A, \mathcal{U}|_A}(\phi|_A) = f_{I,\mathcal{U}}(\phi)$

3. (iteration) if $\{I_i: i \in I\}$ are disjoint sets equipped with ultrafilters $\mathcal{U}_i$, $J = \bigcup I_i$, and $\mathcal{U}$ is an ultrafilter on $I$, then we get an ultrafilter $\mathcal{V}$ on $J$ consisting of the sets whose intersection with each $I_i$ belongs to $\mathcal{U}_i$; then for any $\phi: J \to X$ we have $f_{J,\mathcal{V}}(\phi) = f_{I,\mathcal{U}}(f_{I_i,\mathcal{U}_i}(\phi))$.

I guess that is something like what you're asking for. This is proven in my paper "The variety of CH-algebras", Acta Mathematica Hungarica 69, 221-232 (1995). I guess I wrote this when I was in grad school.