Convergent free sequences in compact spaces

Juhász, I.; Szentmiklóssy, Z.

doi:10.1090/S0002-9939-1992-1137223-8

Convergent free sequences in compact spaces
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by I. Juhász and Z. Szentmiklóssy PDF

Proc. Amer. Math. Soc. 116 (1992), 1153-1160 Request permission

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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