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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

The nonexistence of a continuous surjection from a continuum onto its square

Author: Hidefumi Katsuura
Journal: Proc. Amer. Math. Soc. 111 (1991), 1129-1140
MSC: Primary 54F15; Secondary 54C05, 54D05
MathSciNet review: 1039258
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Abstract: In the late nineteenth century, the Italian mathematician Peano discovered a continuous surjection from $ [0,1]$ onto $ [0,1] \times [0,1]$. This led to the discovery, in the early twentieth century, of the Hahn-Mazurkiewicz Theorem, which states that a continuum (compact, connected metric space) is a continuous image of the unit interval $ [0,1]$ if and only if it is locally connected. (Consequently, honoring Peano's discovery, we call a locally connected continuum a Peano continuum.) Combining this theorem and Urysohn's Lemma, one can prove the existence of a continuous surjection form a Peano continuum $ X$ onto $ X \times X$. This observation motivated the author to consider a continuous surjection from a continuum $ X$ onto $ X \times X$, and led to the discovery of a sufficient condition on a continuum for the nonexistence of such functions.

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

  • [1] David P. Bellamy, The cone over the Cantor set--continuous maps from both directions, Topology Conference Proceeding, Emory University, 1970.
  • [2] Hidefumi Katsuura, Set functions and continuous mappings, Ph. D. dissertation, University of Delaware, 1984.

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Additional Information

PII: S 0002-9939(1991)1039258-1
Article copyright: © Copyright 1991 American Mathematical Society