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Convergent free sequences in compact spaces


Authors: I. Juhász and Z. Szentmiklóssy
Journal: Proc. Amer. Math. Soc. 116 (1992), 1153-1160
MSC: Primary 54D35
DOI: https://doi.org/10.1090/S0002-9939-1992-1137223-8
MathSciNet review: 1137223
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Abstract: We prove that if $ \kappa $ is an uncountable regular cardinal and a compact $ {T_2}$ space $ X$ contains a free sequence of length $ \kappa $, then $ X$ also contains such a sequence that is convergent. This implies that under $ {\text{CH}}$ every nonfirst countable compact $ {T_2}$ space contains a convergent $ {\omega _1}$-sequence and every compact $ {T_2}$ space with a small diagonal is metrizable, thus answering old questions raised by the first author and M. Hušek, respectively.


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DOI: https://doi.org/10.1090/S0002-9939-1992-1137223-8
Article copyright: © Copyright 1992 American Mathematical Society