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Remainders of metrizable and close to metrizable spaces. (English) Zbl 1267.54024
The paper deals with remainders \(Y\) of a Tychonoff space \(X\) which are Lindelöf \(\Sigma\)-spaces or Lindelöf \(p\)-spaces. We mention a few typical results. If \(X\) is a Tychonoff space having a \(\sigma\)-disjoint base \(\mathcal B\) of cardinality \(\leq 2^{\omega}\), then every remainder of \(X\) is a Lindelöf \(\Sigma\)-space. For Lindelöf spaces this assertion is true without cardinal restrictions on \(\mathcal B\). Let \(X\) be a paracompact \(p\)-space. Then every remainder \(Y\) of \(X\) satisfies: (1) for any \(A\subset Y\) of cardinality \(\leq 2^{\omega}\) the closure of \(A\) in \(Y\) is a Lindelöf \(\Sigma\)-space, (2) any closed \(P\subset Y\) with the ccc property is a Lindelöf \(p\)-space, (3) If \(X\) has a separable remainder, then \(Y\) is a Lindelöf \(p\)-space.

MSC:
54D40 Remainders in general topology
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54B05 Subspaces in general topology
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54E35 Metric spaces, metrizability
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