EXTENDING TOPOLOGIES FROM SUBGROUPS TO GROUPS
Bradd Evans Clark, Mary C. Dooley, Victor P. Schneider · 1985
Let G be a group and H a proper subgroup of G. If H is a topological group, a natural question to ask is when can the topology on H be extended to a topology on G that makes G into a topological group? While answering a different question, Sharma in [6] has shown that any topology which makes the center of G into a topological group can be extended in such a way as to make all of G into a topologi cal group. Little else has been written about this question. For the purposes of this paper we shall assume that G is a group and that H is a proper subgroup of G that is also a topological group with topology T. Finally we shall assume that U = {Ua}aEf is a basis for the topology of H at the identity element e. If G is a topological group, then a basis for G can be obtained by mUltiplying each element of U by each element of G. Thus there is a natural way to