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 \(\rho \in N(B)\).

An example of a supernearness space is the following: one takes \(X\) to be the reals, \(B^X\) to be the bounded subsets of the reals (the distance of all pairs of points in \(X\) is bounded by a constant) and \(N(B)\) contain all collections \(\rho\) of subsets of \(X\) such that the topological closure of every \(C \in \rho\) meets the topological closure of \(B\). Leseberg considers the category of all supernearness spaces by defining that an \(SN\)-mapping is a function which is bounded (maps bounded sets to bounded sets) and maps, for each \(B\) the \(\rho \in N(B)\) of the first space to \(\rho \in N(\{f(x): x \in B\})\) in the second space. He then discusses various related notions and relates them to supernearness spaces; in particular he investigates superscreen spaces and strict topological extensions in the context of supernearness spaces. He investigates various categories of spaces related to topology and boundedness and embeds these into the supernearness spaces.