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On a ubiquitous cardinal


Author: Stephen H. Hechler
Journal: Proc. Amer. Math. Soc. 52 (1975), 348-352
MSC: Primary 54A25; Secondary 02K25, 54D99
MathSciNet review: 0380705
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Abstract: We consider five combinatorial or topological structures, each with a certain associated minimal cardinal, and we show that these cardinals are always equal even though it is independent of the axioms of set theory as to just what the value of this common cardinal is. The five structures are the set of functions from $ N$ (the set of natural numbers) into $ N$ under two partial orderings, the rational numbers with respect to closed embeddings into powers of $ N$, a certain subset of $ \beta N - N$ with respect to clopen decompositions, the irrationals with respect to compact decompositions, and a subclass of the Borel sets with respect to closed decompositions. The proofs presented do not require a knowledge of forcing techniques.


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DOI: https://doi.org/10.1090/S0002-9939-1975-0380705-7
Article copyright: © Copyright 1975 American Mathematical Society