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

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.