Universality of the closure space of filters in the algebra of all subsets (Q1100463)

Every closure space, i.e. an ordered pair \({\mathcal X}=<X,{\mathcal F}>\) such that (i) \(\emptyset,X\in {\mathcal F}\subseteq {\mathcal P}(X)\), (ii) \({\mathcal R}\subseteq {\mathcal F}\) implies \(\cap {\mathcal R}\in {\mathcal F}\), defines the proper closure operator \(C_{{\mathcal X}}(Z)=\cap \{Y\in {\mathcal F}:\) \(Y\neq \emptyset\) \& \(Z\subseteq Y\}\), all \(Z\subseteq X\). \(C_{{\mathcal X}}\) is said to be compact if \(C_{{\mathcal X}}(Y)=\cup \{C_{{\mathcal X}}(Y_ f):\) \(Y_ f\) is a finite subset of \(Y\}\). `The closure space of all filters in the lattice of all subsets forms a ``generalized Alexandroff cube'' that is universal for \(T_ 0\)-closure spaces'. From this fact it follows that for every closure space \({\mathcal X}\) the following conditions are equivalent: (i) \({\mathcal X}\) is countable and \(C_{{\mathcal X}}\) is compact, (ii) \({\mathcal X}\) is homeomorphically embeddable in the closure space of the consequence operator of classical propositional logic.