A solution to a problem of Eilenberg concerning dimension lowering mappings

Abstract:

This paper describes a map $f$ from a metric space $X$ (having the same dimension at each of its points) onto a space $Y$ such that $\dim X > \dim Y > 0$ with the property that $\dim K \geqslant \dim f(K)$ for every closed set $K$ contained in $X$. This answers a question posed by Eilenberg in 1936 in The Scottish Book. This question was answered with a more complicated example by Rubin, Schori, and Walsh.