The topological universe of locally precompact semiuniform convergence spaces

The importance of local precompactness has become apparent in theorems of the Ascoli type proved by Bentley and Herrlich (1985) in the realm of filter spaces as well as by Wyler (1984) in the realm of uniform limit spaces. Here local precompactness is studied in the highly convenient setting of sem iuniform convergence spaces which form a common generalization of (symmetric) limit spaces (and thus of symmetric topological spaces), filter spaces and uniform limit spaces (and thus of uniform spaces). It turns out that it leads to a topological universe. Furthermore, locally precompact semiuniform convergence spaces are exactly the precompactly generated semiuniform convergence spaces, and the relationships to locally compact, compact and precompact semiuniform convergence spaces respectively, studied before by Preuك (1995), are clarified.