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Remainders of metrizable spaces and a generalization of Lindelöf $$\Sigma$$-spaces. (English) Zbl 1236.54006
The remainder under any compactification of any metrizable space of weight $$\leq2^\omega$$ is a Lindelöf $$\Sigma$$-space. However there is a locally separable, nowhere locally compact, metrizable space $$X$$ with $$|X|\leq(2^\omega)^+$$ such that no remainder of $$X$$ is a Lindelöf $$\Sigma$$-space. On the other hand, the remainder under any compactification of any metrizable space is a charming space (i.e., a space $$X$$ having a Lindelöf $$\Sigma$$-subspace $$Y$$ such that for each open neighbourhood $$U$$ of $$Y$$ in $$X$$ the subspace $$X\setminus U$$ is a Lindelöf $$\Sigma$$-space). Properties of charming spaces are investigated.

##### MSC:
 54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets) 54B05 Subspaces in general topology 54D20 Noncompact covering properties (paracompact, Lindelöf, etc.) 54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.) 54D40 Remainders in general topology 54H11 Topological groups (topological aspects)
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