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Paracompactness and remainders: around Henriksen-Isbell’s theorem. (English) Zbl 1305.54032
A well-known theorem of Henriksen and Isbell states that a Tychonoff space is of countable type iff some/every remainder of it is Lindelöf. The authors investigate this duality further, especially for spaces that are nowhere locally compact: assume $$B$$ is compact and both $$X$$ and $$B\setminus X$$ are dense in $$B$$ (hence both are nowhere locally compact), if $$X$$ is $$\sigma$$-paraLindelöf and $$B\setminus X$$ is of subcountable type then $$X$$ is Lindelöf and $$B\setminus X$$ is of countable type. Note: (sub)countable type means that every compact sets is contained in a compact set with a countable neighbourhood base (that is a $$G_\delta$$-set). They go on to apply this to provide further conditions on remainders that imply that the space is Lindelöf, or even separable metrizable.
Reviewer: K. P. Hart (Delft)

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