On the lambda function and a quantification of Torhorst Theorem

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