$\omega$-connected continua and Jones’ $K$ function

Abstract:

A continuum $X$ is $\omega$-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 $A \subset X$ then $K\left ( A \right )$ is the intersection of all subcontinua of $X$ that contain $A$ in their interiors. The main theorem shows that if $X$ is an $\omega$-connected continuum and $H$ is a connected nowhere dense subset of $X$, then $K\left ( H \right )$ has a void interior. Several corollaries are established for continua with certain separation properties and a final theorem shows the equivalence of $\omega$-connectedness and $\delta$-connectedness for plane continua.