A new Concept of Convergence Space

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.