On topological groups with a first-countable remainder, II

Abstract

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 2ω. Moreover, the cardinality of G does not exceed 2ω1. 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.