On fiber diameters of continuous maps

摘要: We present a surprisingly short proof that for any continuous map $f : \mathbb{R}^n \rightarrow \mathbb{R}^m$, if $n>m$, then there exists no bound on the diameter of fibers $f$. Moreover, we show when $m=1$, union small $f$ is bounded; $m>1$, need not be bounded. Applications to data analysis are considered.