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
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.