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

DOI:
https://doi.org/10.1090/S0002-9939-1991-1039258-1

MathSciNet review:
1039258

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Abstract: In the late nineteenth century, the Italian mathematician Peano discovered a continuous surjection from onto . 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 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 onto . This observation motivated the author to consider a continuous surjection from a continuum onto , and led to the discovery of a sufficient condition on a continuum for the nonexistence of such functions.

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DOI:
https://doi.org/10.1090/S0002-9939-1991-1039258-1

Article copyright:
© Copyright 1991
American Mathematical Society