Abstract
We prove that if κ \kappa is an uncountable regular cardinal and a compact T 2 {T_2} space X X contains a free sequence of length κ \kappa , then X X also contains such a sequence that is convergent. This implies that under CH {\text {CH}} every nonfirst countable compact T 2 {T_2} space contains a convergent ω 1 {\omega _1} -sequence and every compact T 2 {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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