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.


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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[1] Balog, Z.; Dow, A.; Fremlin, D.; Nyikos, P., Countable tightness and proper forcing, Bull. amer. math. soc., 19, 295-298, (1988) · Zbl 0661.54007
[2] Baumgartner, J.E.; Shelah, S., Remarks on superatomic Boolean algebras, Ann. pure appl. logic, 33, 2, 119-129, (1987) · Zbl 0643.03038
[3] Dow, A., On initially κ-compact spaces, (), 103-108
[4] Dow, A.; Juhász, I., Are initially \(\omega_1\)-compact separable regular spaces compact?, Fund. math., 154, 123-132, (1997) · Zbl 0907.54002
[5] Juhász, I.; Soukup, L., How to force a countably tight, initially \(\omega_1\)-compact and noncompact space?, Topology appl., 69, 3, 227-250, (1996) · Zbl 0853.54006
[6] P. Koszmider, Splitting ultrafilters of the thin-very tall algebra and initially \(\omega_1\)-compactness, preprint, 1995
[7] Koszmider, P., Forcing minimal extensions of Boolean algebras, Trans. amer. math. soc., 351, 3073-3117, (1999) · Zbl 0922.03071
[8] Rabus, M., An \(\omega_2\)-minimal Boolean algebra, Trans. amer. math. soc., 348, 8, 3235-3244, (1996) · Zbl 0859.03026
[9] Todorčevic̀, S., Walks on ordinals and their characteristics, Progr. math., vol. 263, (2007), Birkhäuser · Zbl 1148.03004
[10] Velleman, D., Simplified morasses, J. symbolic logic, 49, 257-271, (1984) · Zbl 0575.03035
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