论文标题
$ \ mathbb {r}^2 $ in no Krasnoselskii编号
No Krasnoselskii number for general sets in $\mathbb{R}^2$
论文作者
论文摘要
For a family $\mathcal{F}$ of sets in $\mathbb{R}^d$, the Krasnoselskii number of $\mathcal{F}$ is the smallest $m$ such that for any $S \in \mathcal{F}$, if every $m$ points of $S$ are visible from a common point in $S$, then any finite subset of $S$ is从一个点可见。 35年前，彼得森询问是否存在$ \ mathbb {r}^d $中的一般集的Krasnoselskii编号。不包括具有强大拓扑限制的特殊情况的结果，最著名的结果是由于布雷恩（Breen）所致，他表明，如果存在这样的krasnoselskii编号，则存在于$ \ mathbb {r}^2 $中，那么它大于$ 8 $。 在本文中，我们以否定的方式回答了彼得森的问题，表明在$ \ mathbb {r}^2 $中没有krasnoselskii编号。证明是非构造性的，并使用递限诱导和井顺序定理。 此外，我们考虑了通过长度$ \ leq n $的多边形路径的可见度来考虑Krasnoselskii的数字，为此，Magazanik和Perles证明了Krasnoselskii定理的类似物。我们通过明确的结构显示，对于任何$ n \ geq 2 $，在$ \ mathbb {r}^2 $中，对于通过长度$ \ leq n $的可见度而言，通用集的家族都没有Krasnoselskii编号。 （在这里反例是线段的有限工会。）
For a family $\mathcal{F}$ of sets in $\mathbb{R}^d$, the Krasnoselskii number of $\mathcal{F}$ is the smallest $m$ such that for any $S \in \mathcal{F}$, if every $m$ points of $S$ are visible from a common point in $S$, then any finite subset of $S$ is visible from a single point. More than 35 years ago, Peterson asked whether there exists a Krasnoselskii number for general sets in $\mathbb{R}^d$. Excluding results for special cases of sets with strong topological restrictions, the best known result is due to Breen, who showed that if such a Krasnoselskii number in $\mathbb{R}^2$ exists, then it is larger than $8$. In this paper we answer Peterson's question in the negative by showing that there is no Krasnoselskii number for the family of all sets in $\mathbb{R}^2$. The proof is non-constructive, and uses transfinite induction and the well ordering theorem. In addition, we consider Krasnoselskii numbers with respect to visibility through polygonal paths of length $ \leq n$, for which an analogue of Krasnoselskii's theorem was proved by Magazanik and Perles. We show, by an explicit construction, that for any $n \geq 2$, there is no Krasnoselskii number for the family of general sets in $\mathbb{R}^2$ with respect to visibility through paths of length $\leq n$. (Here the counterexamples are finite unions of line segments.)
