A note on topological groups and their remainders

In this note we first give a summary that on property of a remainder of a nonlocally compact topological group G in a compactification bG makes the remainder and the topological group G all separable and metrizable. If a non-locally compact topological group G has a compactification bG such that the remainder bG \ G of G belongs to P, then G and bG \ G are separable and metrizable, where P is a class of spaces which satisfies the following conditions: Some known conclusions on topological groups and their remainders can be obtained from this conclusion. As a corollary, we have that if a non-locally compact topological group G has a compactification bG such that compact subsets of bG \ G are G δ -sets in a uniform way (i.e., bG \ G is CSS), then G and bG \ G are separable and metrizable spaces. In the last part of this note, we prove that if a non-locally compact topological group G has a compactification bG such that the remainder bG \ G has a point-countable weak base and has a dense subset D such that every point of the set D has countable pseudo-character in the remainder bG \ G (or the subspace D has countable π-character), then G and bG\G are both separable and metrizable.