论文标题
关于lambda函数和torhorst定理的量化
On Lambda Function and a Quantification of Torhorst Theorem
论文作者
论文摘要
对于任何紧凑的$ k \ subset \ hat {\ mathbb {c}} $，我们将图$λ_k：\ hat {\ hat {\ mathbb {c}} \ rightarrow \ rightarrow \ rightarrow \ mathbb {n} \ cup \ cup \ cup \ {\ infty \ fiftty \}如果$λ_k（x）\ equiv0 $。我们建立了确定lambda函数的基本方法$λ_K$，用于特定的compacta $ k \ subset \ hat {\ mathbb {c}} $，包括lambda函数的胶粘引理和一些不等式。这些不等式之一来自平面compactum $ k $的拓扑难度与属于$ \ hat {\ hat {\ mathbb {c}} \ setMinus k $的组件的边界的相互作用。它概括并量化Torhorst定理的结果，这是平面拓扑的基本结果。我们还发现这种不平等实际上是平等的三个条件。在这些条件之一下，这种平等为Whyburn定理提供了定量版本，这是与Torhorst定理的部分交谈。
To any compact $K\subset\hat{\mathbb{C}}$ we associate a map $λ_K: \hat{\mathbb{C}}\rightarrow\mathbb{N}\cup\{\infty\}$ -- the lambda function of $K$ -- such that a planar continuum $K$ is locally connected if and only if $Λ_K(x)\equiv0$. We establish basic methods of determining the lambda function $λ_K$ for specific compacta $K\subset\hat{\mathbb{C}}$, including a gluing lemma for lambda functions and some inequalities. One of these inequalities comes from an interplay between the topological difficulty of a planar compactum $K$ and that of a sub-compactum $L\subset K$, lying on the boundary of a component of $\hat{\mathbb{C}}\setminus K$. It generalizes and quantifies the result of Torhorst Theorem, a fundamental result from plane topology. We also find three conditions under which this inequality is actually an equality. Under one of these conditions, this equality provides a quantitative version for Whyburn's Theorem, which is a partial converse to Torhorst Theorem.
