On the lambda function and a quantification of Torhorst Theorem

Abstract

For any compact K⊂Cˆ we define a map λK:Cˆ→N∪{∞}, called the lambda function of K. The lambda function of a continuum K has the property that λK(x)=0 for all x∈Cˆ if and only if K is locally connected. We establish several inequalities and a gluing lemma for the lambda functions. These inequalities reflect the relationship between the topology of K and the complexity that may be involved in the boundary of a component of Cˆ∖K. One of them, called the lambda inequality, generalizes and quantifies the Torhorst Theorem. We also find three sufficient conditions under each of which the lambda inequality becomes an equality. These results cover Whyburn's Theorem on the partial converse to the Torhorst Theorem. The gluing lemma for the lambda functions is of research interest in the study of point set topology and of polynomial Julia sets as well.