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2 Fundamenta Mathematicae

2 Arhangel'skii A.

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Remainders of metrizable and close to metrizable spaces

100%

Arhangel'skii A.

nr 1

71-81

We continue the study of remainders of metrizable spaces, expanding and applying results obtained in [Fund. Math. 215 (2011)]. Some new facts are established. In particular, the closure of any countable subset in the remainder of a metrizable space is a Lindelöf p-space. Hence, if a remainder of a metrizable space is separable, then this remainder is a Lindelöf p-space. If the density of a remainder Y of a metrizable space does not exceed $2^{ω}$, then Y is a Lindelöf Σ-space. We also show that many of the theorems on remainders of metrizable spaces can be extended to paracompact p-spaces or to spaces with a σ-disjoint base. We also extend to remainders of metrizable spaces the well known theorem on metrizability of compacta with a point-countable base.

A study of remainders of topological groups

100%

Arhangel'skii A.

Fundamenta Mathematicae

2009

tom 203

nr 2

165-178

Some duality theorems relating properties of topological groups to properties of their remainders are established. It is shown that no Dowker space can be a remainder of a topological group. Perfect normality of a remainder of a topological group is consistently equivalent to hereditary Lindelöfness of this remainder. No L-space can be a remainder of a non-locally compact topological group. Normality is equivalent to collectionwise normality for remainders of topological groups. If a non-locally compact topological group G has a hereditarily Lindelöf remainder, then G is separable and metrizable. We also present several other criteria for a topological group G to be separable and metrizable. Two of them are of general nature and depend heavily on a new criterion for Lindelöfness of a topological group in terms of remainders. One of them generalizes a theorem of the author [Topology Appl. 150 (2005)] as follows: a topological group G is separable and metrizable if and only if some remainder of G has locally a $G_δ$-diagonal. We also study how close are the topological properties of topological groups that have homeomorphic remainders.