On Finite-to-One Maps

[摘要] Let $fcolon Xo Y$ be a $sigma$-perfect $k$-dimensional surjectivemap of metrizable spaces such that $dim Yleq m$. It is shown thatfor every positive integer $p$ with $ pleq m+k+1$ there exists adense $G_{delta}$-subset ${mathcal H}(k,m,p)$ of $C(X,uin^{k+p})$with the source limitation topology such that each fiber of$friangle g$, $gin{mathcal H}(k,m,p)$, contains at most$max{k+m-p+2,1}$ points. This resultprovides a proof the following conjectures ofS. Bogatyi, V. Fedorchuk and J. van Mill.Let $fcolon Xo Y$ be a $k$-dimensional map between compactmetric spaces with $dim Yleq m$. Then:egin{inparaenum}[m(1)]item there exists a map$hcolon Xouin^{m+2k}$ such that $friangle hcolon XoYimesuin^{m+2k}$ is 2-to-one provided $kgeq 1$;item there exists amap $hcolon Xouin^{m+k+1}$ such that $friangle hcolon XoYimesuin^{m+k+1}$ is $(k+1)$-to-one.end{inparaenum}

[学科分类] 数学（综合）

[关键词] finite-to-one maps;dimension;set-valued maps