On topological groups with a first-countable remainder, II
On topological groups with a first-countable remainder, II
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DOI：
10.1016/j.topol.2015.09.015
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发表时间：
2015-11
影响因子：
0.6
通讯作者：
A. Arhangel'skii;J. Mill
中科院分区：
文献类型：
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作者：
A. Arhangel'skii;J. Mill
我们对具有第一可数余数 Y 的任意非局部紧拓扑群 G 的基数不变量进行估计。我们证明 G 的权重和 Y 的基数不超过 2 ω。此外，G 的基数不超过 2 ω 1。这些界限是最好的可能，正如单个拓扑群 G 所证明的那样。我们还证明了每个具有第一可数余数的预紧拓扑群是可分离和可度量的。众所周知，在马丁公理和连续统假设的否定下，每个具有第一可数余数的σ-紧拓扑群都是可度量的。我们证明，在连续统假设下，存在一个不可度量且具有第一可数余数的可数拓扑群的例子。因此，对于可数群，第一可数余数的存在是否等同于可度量的问题是不可判定的。
We establish estimates on cardinal invariants of an arbitrary non-locally compact topological group G with a first-countable remainder Y. We show that the weight of G and the cardinality of Y do not exceed 2 ω. Moreover, the cardinality of G does not exceed 2 ω 1. These bounds are best possible as witnessed by a single topological group G. We also prove that every precompact topological group with a first-countable remainder is separable and metrizable. It is known that under Martin's Axiom and the negation of the Continuum Hypothesis, every σ-compact topological group with a first-countable remainder is metrizable. We show that under the Continuum Hypothesis, there is an example of a countable topological group which is not metrizable and has a first-countable remainder. Hence for countable groups, the question of whether the existence of a first-countable remainder is equivalent to being metrizable, is undecidable.