**MSC:**- 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

in

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.