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A first countable, initially \(\omega _1\)-compact but non-compact space. (English) Zbl 1168.54003

It is known that under CH an initially \(\omega_1\)-compact \(T_3\) space of countable tightness is compact. In this paper, by forcing, the authors construct a locally compact, first countable, zero-dimensional, normal, initially \(\omega_1\)-compact, non-compact space \(X\) of cardinality \(\omega_2\). An interesting consequence is that the Alexandroff one-point compactification of \(X\) has no non-trivial converging \(\omega_1\)-sequences.

MSC:

54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
03E35 Consistency and independence results
54A35 Consistency and independence results in general topology
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