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In the realm of Bounded Topology we now consider supernearness spaces as a common generalization of various kinds of topological structures. Among them the so-called Lodato spaces are of significant interest. In one direction they are standing in one-to-one correspondence to some kind of topological extensions. This last statement also holds for contiguity spaces in the sense of Ivanova and Ivanov, respectively and moreover for bunch-determined nearness spaces as Bentley has shown in the past. Further, Do?tch?nov proved that the compactly determined Hausdorff extensions of a given topological space are closely connected with a class of supertopologies which he called b-supertopologies. Now, the new class of supernearness spaces—called paranearness spaces—generalize all of them, and moreover its subclass of clan spaces is in one-to-one correspondence to a certain kind of symmetric strict topological extension. This is leading us to one theorem which generalize all former mentioned.

The notion of b-convergence is presented for studying preuniform convergence spaces in the sense of Preuß (1993) and set-convergence spaces introduced by Wyler in 1989 from a common point of view. The well-known supertopologies as defined by Doitchinov in 1964 and also the filtermerotopies in the sense of Kat´etov (1965) can be integrated as well. Even the grill-defined presupernear operators, introduced by the author (1999) are contained in this new broader concept. Moreover, we discuss all the properties for describing categories in the realm of Convenient Topology, especially the properties of being cartesian closed or extensional.