ω-Connected Continua and Jones' K Function

Abstract

continuum X is u-connected if for every pair of points x. y of X, there exists an irreducible subcontinuum of X from x to y that is decomposable. If c X then K(A) is the intersection of all subcontinua of X that contain in their interiors. The main theorem shows that if X is an u-connected continuum and H is a connected nowhere dense subset of X. then K(H) has a void interior. Several corollaries are established for continua with certain separation properties and a final theorem shows the equivalence of «-connectedness and S-connectedness for plane continua. Let A be a compact connected metric space (continuum). If for every pair of distinct points x, y of X there exists a subcontinuum / of X irreducible between x and y such that; (1) / is decomposable, then X is u-connected (3); (2) / contains no indecomposable subcontinuum with nonvoid interior relative to /, then X is X-con- nected (7); (3) / is hereditarily decomposable, then / is 8-connected (7); (4) / is an arc, then / is a-connected (arcwise connected). The first three properties are generaliza- tions of arcwise connectedness and it is clear that a-connectedness implies 5-con- nectedness which implies A-connectedness which implies co-connectedness. Also if c X let K(A) be the intersection of all subcontinua of X that contain in their interiors relative to X. This concept was introduced by F. B. Jones in (9, Theorem 2).