Gerhard Preuß
Was ist der geeignete Rahmen zur Behandlung topologischer Probleme?
The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq.51, 141-158(1997)
54A05 Topological spaces and generalizations (closure spaces, etc.)
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, etc.)
54C35 Function spaces, See also {46Exx, 58D15}
54D45 Local compactness, $sigma$-compactness
54D50 $k$-spaces
54E15 Uniform structures and generalizations
Abstract: At first several desirable properties of a concept of `space'
Topology are considered. Unfortunately, the usual concept of topological space
does not fulfill any of them. Also uniform spaces do not behave much
better. Thus, some improvements of the concept of space are discussed such as
limit spaces or uniform limit spaces. But even these spaces do not have all
the mentioned properties. Then a satisfactory and simple solution is presented
by introducing semiuniform convergence spaces, whose systematic study has been
begun by the author [29] in 1995.
Keywords: Topological spaces, uniform spaces, limit spaces, uniform limit spaces, natural function spaces, hereditary quotients, productivity of quotients, local (pre-) compactness, semiuniform convergence spaces.