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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References:

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