Every topological space admits
at least one compatible transitive quasi-uniformity, and among the compatible
transitive quasi-uniformities there is always a finest one. It is shown that a
topological space is compact if and only if its finest transitive quasi-uniformity
is precompact. A TI space X whose fine quasi-uniformity is transitive is
quasi-metrizable if and only if X has a σ − Q-base. The upper semi-continuous
quasi-uniformity of any topological space is transitive and countably precompact and
a space X is almost realcompact if and only if it has a compatible almost complete
countably precompact transitive quasi-uniformity. Consequently a space is almost
realcompact if and only if its upper semi-continuous quasi-uniformity is almost
complete. The latter result is the natural analogue of the result of Shirota
that a space X is realcompact if and only if it is complete in the structure
C(X).
