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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A topological version of $ \diamondsuit $


Author: John Ginsburg
Journal: Proc. Amer. Math. Soc. 65 (1977), 142-144
MSC: Primary 04A20; Secondary 54A20, 54A25
DOI: https://doi.org/10.1090/S0002-9939-1977-0441737-5
MathSciNet review: 0441737
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Abstract: The set-theoretic principle $ \diamondsuit $ is shown to be equivalent to the existence of universal $ {\omega _1}$-sequences in certain topological spaces. An $ {\omega _1}$-sequence $ ({x_\alpha }:\alpha \in {\omega _1})$ in a space X is said to be universal in X if for every point $ p \in X$ there is a stationary set $ S \subseteq {\omega _1}$ so that the net $ ({x_\alpha }:\alpha \in S)$ converges to p. It is shown that the existence of universal $ {\omega _1}$-sequences in spaces of weight $ \leqslant c$ is equivalent to $ \diamondsuit $.


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DOI: https://doi.org/10.1090/S0002-9939-1977-0441737-5
Keywords: $ \diamondsuit $, stationary sets, convergence, universal $ {\omega _1}$-sequence
Article copyright: © Copyright 1977 American Mathematical Society