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]×[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×X. This observation motivated the author to consider a continuous surjection from a continuum X onto X×X, and led to the discovery of a sufficient condition on a continuum for the nonexistence of such functions