Abstract
Let P be the class of all spaces Y satisfying:(1)Every compact subset of Y is a Gδ-set;(2)If Y is countably compact, then it is locally compact;(3)Every closed Lindelöf p-subspace of Y is metrizable. We show that if X is a nowhere locally compact compactly-fibered coset space and bX is a compactification of X such that the remainder bX∖X of X is in the class P, then bX∖X and X are separable metrizable spaces. Since spaces having a point countable base or a Gδ-diagonal are in the class P, this generalizes results of Arhangel'skii. We obtain a number of related results, and also consider when a remainder of a space is a D-space.
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