关于正则分支映射
On Regularly Branched Maps
摘要：
设$f\colon X\to Y$是有限维可度量化空间与$p\geq 1$之间的完美映射.证明了从$X$到$\R^p$的所有具有源限制拓扑的有界映射的空间$C^*(X,\R^p)$包含由$f$ -正则分支映射构成的稠密$G_δ$ -子集.这里，如果对于每个$n\geq 1$，集合$\{z\in Y\times\R^p: |(f\times g)^{-1}(z)|\geq n\}$的维数是$\leq n\cdot\big(\dim f+\dim Y\big)-(n-1)\cdot\big(p+\dim Y\big)$，则映射$g\colon X\to\R^p$是$f$ -规则分支的。这是正则分支映射上的Hurewicz定理的参数版本。
英文摘要：
Let $f\colon X\to Y$ be a perfect map between finite-dimensional metrizable spaces and $p\geq 1$. It is shown that the space $C^*(X,\R^p)$ of all bounded maps from $X$ into $\R^p$ with the source limitation topology contains a dense $G_δ$-subset consisting of $f$-regularly branched maps. Here, a map $g\colon X\to\R^p$ is $f$-regularly branched if, for every $n\geq 1$, the dimension of the set $\{z\in Y\times\R^p: |(f\times g)^{-1}(z)|\geq n\}$ is $\leq n\cdot\big(\dim f+\dim Y\big)-(n-1)\cdot\big(p+\dim Y\big)$. This is a parametric version of the Hurewicz theorem on regularly branched maps.
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