Abstract.

For a uniform space uX the concept of Cu-embedding (C ∗ u-embedding) in some uniform space is introduced. An analogue of Urysohn’s Theorem is proved and it is established, that uX is C ∗ u 􀀀embedded in the Wallman 􀀀like compactification uX, and any compactification of uX in which uX is C ∗ u 􀀀embedded, must be uX. A uniformly realcompact space is determined. It is proved, that uX is Cu􀀀embedded in the Wallman realcompactification uX, and any uniform realcompactification in which uX is Cu􀀀embedded, is uX.

Keywords:

u-open (closed) sets, zu-filter, zu-ultrafilter, coz-mapping, u-continuous function, Wallman b-like compactification, Wallman realcompactification.