Improved nearness research. III. (English) Zbl 1266.54005
The author described in his monograph [Strict topological extensions and power-objects for b-convergence. Saarbrücken: LAP Lambert Academic Publishing (2015; Zbl 1345.54004)] bounded spaces which are given by a base set \(X\) and a class of bounded sets \(B^X\) which contains all singleton sets and which is closed under subsets. Now a triple \((X,B^X,N)\) is called a supernearness space where \(N\) assigns to every bounded set \(B \in B^X\) a subset of the power set of the power set of \(X\) which contains all collections of sets which converge in some way to a point in \(B\). The formal axioms are the following:
- \((sn_1)\)
- If \(B \in B^X\) and \(\rho_1 \in N(B)\) and every set in \(\rho_2\) is a superset of a set in \(\rho_1\) then \(\rho_2 \in N(B)\);
- \((sn_2)\)
- For every \(B \in B^X\), \(N(B)\) is not empty and \(B^X\) is not contained in \(N(B)\);
- \((sn_3)\)
- \(N(\emptyset)\) contains only the empty collection of sets;
- \((sn_4)\)
- \(N(\{x\})\) contains \(\{\{x\}\}\);
- \((sn_5)\)
- If \(B,C \in B^X\) and \(B \subseteq C\) then \(N(B) \subseteq N(C)\);
- \((sn_6)\)
- If \(B \in B^X\) and \(\{F_1 \cup F_2: F_1 \in \rho_1, F_2 \in \rho_2\} \in N(B)\) then at least one of \(\rho_1,\rho_2\) is in \(N(B)\);
- \((sn_7)\)
- If \(B \in B^X\), \(\rho\) is a subset of the power set of \(X\) and \(\{\{x \in X: \{F\} \in N(x)\}: F \in \rho\} \in N(B)\) then \(\rho \in N(B)\).
Reviewer: Frank Stephan (Singapore)
MSC:
54A05 | Topological spaces and generalizations (closure spaces, etc.) |
54E05 | Proximity structures and generalizations |
18B30 | Categories of topological spaces and continuous mappings (MSC2010) |
54B30 | Categorical methods in general topology |
54D80 | Special constructions of topological spaces (spaces of ultrafilters, etc.) |