Prove the following: theorem: every topological group is
completely regular. Proof. Let V0 be a neighborhood of the identity
elemetn e, in the topological group G. In general, coose Vn to be a
neighborhood of e such that Vn.VncVn-1. Consider the set of all
dyadic rationals p, that is all ratinal number of the form k/sn,
with k and n inegers. FOr each dyadic rational p in (0,1], define
an open set U(p) inductively as foloows: U(1)=V0 and