The nonexistence of a continuous surjection from a continuum onto its square
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- by Hidefumi Katsuura PDF
- Proc. Amer. Math. Soc. 111 (1991), 1129-1140 Request permission
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
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David P. Bellamy, The cone over the Cantor set—continuous maps from both directions, Topology Conference Proceeding, Emory University, 1970.
Hidefumi Katsuura, Set functions and continuous mappings, Ph. D. dissertation, University of Delaware, 1984.
Additional Information
- © Copyright 1991 American Mathematical Society
- 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