A note on topological groups and their remainders

作者：Peng Liang Xue^*; He Yu Feng

来源：Czechoslovak Mathematical Journal, 2012, 62(1): 197-214.

DOI：10.1007/s10587-012-0006-9

• topological group
• remainder
• compactification
• metrizable space
• weak base

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:
(1)if X is an element of P, then every compact subset of the space X is a G (delta) -set of X
(2) if X is an element of P and X is not locally compact, then X is not locally countably compact
(3) if X is an element of P and X is a Lindelof p-space, then X is metrizable.
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 (delta) -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 pi-character), then G and bG\G are both separable and metrizable.

• 出版日期2012-3
• 单位北京工业大学