On Lambda Function and a Quantification of Torhorst Theorem

Abstract

To any compact $K\subset \hat{\mathbb{C}}$ we associate a map $\hat{I}{»}_K:\hat{\mathbb{C}}\to \mathbb{N}\cup \{\infty \}$ -- the lambda function of $K$ -- such that a planar continuum $K$ is locally connected if and only if $\hat{I}{›}_K(x)\equiv 0$. We establish basic methods of determining the lambda function $\hat{I}{»}_K$ 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.