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Remainders in compactifications and generalized metrizability properties. (English) Zbl 1075.54012
The starting point for this article is the classical result of Henriksen and Isbell that a space \(X\) is of countable type if and only if the remainder in any (or in some) compactification of \(X\) is Lindelöf. This result implies for example that every remainder of a metrizable space is Lindelöf. A space \(X\) has property \(\mathcal P\) at infinity if some remainder of \(X\) has the property \(\mathcal P\). Hence each metrizable space is Lindelöf at infinity. The author considers spaces whose remainders are, in some sense, close to being metrizable and presents interesting results about them. He proves for example that every remainder of a Lindelöf \(p\)-space is a Lindelöf \(p\)-space. This does not generalize to paracompact \(p\)-spaces. Interestingly, for topological groups, it does generalize. Several open problems are posed. One of them is to characterize the nowhere locally compact topological groups that have a metrizable remainder.

54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54B05 Subspaces in general topology
54D40 Remainders in general topology
54H11 Topological groups (topological aspects)
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