A note on Jones’ function $K$

Abstract:

For each point $x$ of a continuum $M$, F. B. Jones [5, Theorem 2] defines $K(x)$ to be the closed set consisting of all points $y$ of $M$ such that $M$ is not aposyndetic at $x$ with respect to $y$. Suppose $M$ is a plane continuum and for any positive real number $\epsilon$ there are at most a finite number of complementary domains of $M$ of diameter greater than $\epsilon$. In this paper it is proved that for each point $x$ of $M$, the set $K(x)$ is connected.