Improved nearness research. II (Q2871484)

This is the second out of a series of three papers on supernearness operators.NEWLINENEWLINEThe author investigates bounded sets in a topological space in connection with supernearness operators. Such interactions are quite common in topology, in particular the notion of compactness originates from combining the notion of boundedness with the notion of closedness for subsets of the real line or plane or related spaces. While nearness and neighbourhood relations have been studied abundantly, the notion of boundedness is less popular in topology. The author formalises boundedness by studying the collection \(\mathcal B\) of bounded sets; such a collection \(\mathcal B\) must contain every singleton and is closed under subset-formation and finite unions.NEWLINENEWLINEFurthermore, a supernearness operator \(N\) assigns to every bounded set \(B \in \mathcal B\) a set of possible collections of near sets. For the empty set, each such collection is empty. For having non-triviality, \(\mathcal B\) itself cannot be a collection of near sets of any \(B \in \mathcal B\). For a singleton \(x\), \(\{\{x\}\}\) is one possible collection of near sets. If \(\rho\) is a possible collection of near sets for \(B\) and \(\sigma\) satisfies \(\forall F \in \sigma \exists G \in \rho\,[G \subseteq F]\) then \(\sigma\) is also a possible collection of near sets for \(B\). If \(B \subseteq C\) then every possible collection of near sets of \(B\) is also a possible collection of near sets of \(C\). Given two possible collections \(\rho,\sigma\) of near sets, one defines \(\rho \vee \sigma\) as the collection of all sets \(F \cup G\) with \(F \in \rho\) and \(G \in \sigma\); if now \(\rho \vee \sigma\) is a possible collection of near sets of \(B\) then so is either \(\rho\) or \(\sigma\). A special importance have possible collections of near sets which are singletons, they define a closure operation \(cl(F) = \{x: \{F\}\) is a possible collection of near sets for \(\{x\}\}\). The last axiom of the supernearness space says that if \(\{cl(F): F \in \rho\}\) is a possible collection of near sets of a set \(B\) then so is \(\rho\) itself.NEWLINENEWLINEIn addition to bounded sets and supernearness operators, Leseberg also defines the notion of a supernearness function. A function \(f:X \rightarrow X'\) from one supernearness space \((X,{\mathcal B},N)\) to another such space \((X',{\mathcal B}',N')\) where \({\mathcal B},{\mathcal B}'\) are the collection of bounded sets and \(N,N'\) the supernearness operators is called a supernearness function iff \(B \in {\mathcal B}\) and \(\rho \in N(B)\) imply that \(f[B] \in {\mathcal B}'\) and \(\{f[F]: F \in \rho\} \in N(f[B])\) where \(f[A] = \{f(a): a \in A\}\) is the image of the set \(A\) and \(N(B)\) is the set of all possible collections of near sets of \(B\).NEWLINENEWLINEA natural example of a supernearness space is a locally compact but not compact space \(X\), the collection \(\mathcal B\) of all sets \(B \subseteq X\) such that \(B\) is contained in a compact subset of \(X\) and, for \(B \in \mathcal B\), \(N(B)\) consists of all \(\rho\) which contain only sets whose topological closure intersects with \(B\). If \(f\) is now a homeophism from \(X\) to \(X\) then it is also an example of a supernearness map. Note that there are many choices of such \(X\), the most prominent one would be the set of all real numbers with its natural topology.NEWLINENEWLINEStarting from this category SN of supernearness spaces and supernearness functions, Leseberg develops a comprehensive theory of its natural subcategories and investigates among other topics the inclusion-relations between these subcategories. In the later part of his paper, he studies topological extensions of such spaces.