具有维数受限纤维的映射
Maps with dimensionally restricted fibers
摘要：
证明了：若$f\colon X\to Y$是度量空间之间的闭满射，使得每个纤维$f^{-1}(y)$都属于一类空间$\mathrm S$，则存在$F_σ$ -集$A\subset X$，使得对所有$y\in Y$，$A\in\mathrm S$和$\dim f^{-1}(y)\backslash A=0$.这里，$\mathrm S$可以是以下类别之一：（i）对于某些$CW$ -复$K$的$\{M:\mathrm{e-dim}M\leq K\}$;（ii）$C$ -空间;（iii）弱无限维空间。我们还确定，如果$\mathrm S=\{M:\dim M\leq n\}$，那么几乎所有的$g\in C(X,\mathbb I^{n+1})$的$\dim f\triangle g\leq 0$。
英文摘要：
We prove that if $f\colon X\to Y$ is a closed surjective map between metric spaces such that every fiber $f^{-1}(y)$ belongs to a class of space $\mathrm S$, then there exists an $F_σ$-set $A\subset X$ such that $A\in\mathrm S$ and $\dim f^{-1}(y)\backslash A=0$ for all $y\in Y$. Here, $\mathrm S$ can be one of the following classes: (i) $\{M:\mathrm{e-dim}M\leq K\}$ for some $CW$-complex $K$; (ii) $C$-spaces; (iii) weakly infinite-dimensional spaces. We also establish that if $\mathrm S=\{M:\dim M\leq n\}$, then $\dim f\triangle g\leq 0$ for almost all $g\in C(X,\mathbb I^{n+1})$.
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