DIEUDONN ÉCOMPLETE BISPACES

A bispace is called Dieudonn ´e complete if it admits a compatible bicomplete quasi- uniformity.We characterize pairwise Tychono .Dieudonn ´e complete bispaces in terms of their situation in the canonical bitopological compacti .cation and,from this result,derive a characterization of those T 0 topological spaces that admit a bicomplete quasi-uniformity. These characterizations are given in terms of bispaces which are 2perfect pre-images of bicompletely quasi-metrizable bispaces,which,in turn,are characterized in terms of the notion of an M -bispace.Using an example of R.Fox,we show that there exists a non quasi- metrizable bispace (X,P,Q)which has a Q ×P -G . -diagonal and is the 2perfect pre-image of a quasi-metrizable bispace,in contrast with the analogous metric topological theorem.