𝜔-connected continua and Jones’ 𝐾 function

Abstract

A continuum X X is ω \omega -connected if for every pair of points x x , y y of X X , there exists an irreducible subcontinuum of X X from x x to y y that is decomposable. If A ⊂ X A \subset X then K ( A ) K\left ( A \right ) is the intersection of all subcontinua of X X that contain A A in their interiors. The main theorem shows that if X X is an ω \omega -connected continuum and H H is a connected nowhere dense subset of X X , then K ( H ) K\left ( H \right ) has a void interior. Several corollaries are established for continua with certain separation properties and a final theorem shows the equivalence of ω \omega -connectedness and δ \delta -connectedness for plane continua.