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Images of not Lindelöf spaces and their squares. (English) Zbl 0861.54002

Summary: If \(X\) is not Lindelöf and either \(X\) is locally compact or some closed \(A\) in \(X\) satisfies \(d(A)<L(A)\), then \(X^2\) has a not realcompact continuous image. If \(X\) is locally Lindelöf but not Lindelöf or if \(X\) is not linearly Lindelöf then either \(X^2\) has a not realcompact continuous image or some closed subspace of \(X\) maps continuously onto a regular uncountable cardinal. If \(X\) is either locally Lindelöf or not linearly Lindelöf and \(X\) is normal, then \(X^2\) has a not realcompact continuous image.

MSC:

54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54D60 Realcompactness and realcompactification
54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
54C05 Continuous maps
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