On Uniformities and Uniformly continuous functions in Topological groups
by
Siofilisi Hingano
Victoria University of Wellington

It is well known that there are two natural uniformities in a topological group: the left and the right uniformities. A topological group G is called SIN (from small Invariant Neighbourhoods) if these two uniformities coincide, and FSIN (Functionally SIN) if the class of left uniformly continuous and right uniformly continuous real valued functions on G coincide. Plainly every SIN groups is FSIN, but it still unknown if the converse holds true.

The problem on the coincidence of the two classes was suggested some years ago by Gerald Itzkowitz. It was answered in the affirmative for some particular classes of topological groups, including metrizable groups, locally compact groups and some other classes. I will be talking of some recent new advances.

Date received: October 11, 2000