Approach spaces, limit tower spaces, and probabilistic convergence spaces (Q1360359)

It is shown that the category CAP of convergence approach spaces is isomorphic to the category LTS of limit tower spaces. CAP has, as objects, pairs \((X,\lambda)\), where \(X\) is a set and \(\lambda:F(X)=\) (the set of filters on \(X)\to[0, \infty]^X\) satisfies (1) \(\lambda(\dot x)(x)=0\), where \(\dot x\) is the filter generated by \(\{x\}\), and for all \(F,G\in F(X)\), (2) \(\lambda(F\cap G)=(\lambda F)\vee (\lambda G)\), and (3) \(\lambda G\leq\lambda F\) if \(F\subset G\), and has as morphisms the contraction maps. LTS has, as objects, pairs \((X,\overline{p})\), where \(X\) is a set and \(\overline{p}\) is a family \(\{P_\varepsilon\}\), \(\varepsilon\in [0,\infty]\), of limit structures \(P_\varepsilon\) (i.e. of functions \(P=P_\varepsilon: F(X)\to 2^X\) satisfying \(x\in P(\dot x)\), \(F\subset G\) implies \(P(F)\subset P(G)\), and \(x\in P(F)\cap P(G)\) implies \(x\in P(F\cap G))\) satisfying \(\varepsilon\leq \gamma\) implies \(P_\gamma\leq P_\varepsilon\), \(P_\infty\) is the indiscrete topology, and \(P_\varepsilon= \sup\{P_\gamma\mid \varepsilon<\gamma\}\), and has, as morphisms, appropriately defined continuous maps. The isomorphism between CAP and LTS is explicitly described and its restriction to various subcategories of CAP, such as pre-, pseudo-, and approach spaces, is shown to induce an isomorphism to appropriately described subcategories of LTS, which, in turn, is shown to be isomorphic to the category PTS of probabilistic topological spaces. The paper concludes with a convenient table showing the isomorphic correspondences among the various subcategories of CAP, LTS, and PTS.