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Proceedings of the American Mathematical Society

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

 
 

 

Rigid pairs of long arcs


Author: Gary G. Miller
Journal: Proc. Amer. Math. Soc. 31 (1972), 591-594
DOI: https://doi.org/10.1090/S0002-9939-1972-0290336-2
MathSciNet review: 0290336
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Abstract | References | Additional Information

Abstract: Arcs (linearly ordered continua) $ A$ and $ B$ are constructed such that every map from $ A$ to $ B$ and every map from $ B$ to $ A$ is constant. The Generalized Continuum Hypothesis is sufficient for the existence of two such arcs each of cardinality $ {2^\aleph }$ for each uncountable cardinal $ \aleph $.


References [Enhancements On Off] (What's this?)

  • [1] G. Birkhoff, Lattice theory, 3rd ed., Amer. Math. Soc. Colloq. Publ., vol. 25, Amer. Math. Soc., Providence, R.I., 1967. MR 37 #2638. MR 0227053 (37:2638)
  • [2] M. A. Maurice, Compact ordered spaces, Mathematical Centre Tracts, 6, Mathematisch Centrum, Amsterdam, 1964. MR 36 #3318. MR 0220252 (36:3318)
  • [3] G. Miller Jumps and gaps in lexicographic products (to appear).
  • [4] V. Novák, On the lexicographic dimension of linearly ordered sets, Fund. Math. 56 (1964), 9-20. MR 34 #7412. MR 0207597 (34:7412)


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0290336-2
Keywords: Arcs, linearly ordered continua, continuous functions, Urysohn's lemma, lexicographic product, ordinal power, rigid pairs of spaces
Article copyright: © Copyright 1972 American Mathematical Society

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