aposyndetic

A connected space X is aposyndetic at H with respect to K if there is a closed connected subset of X with H in its interior and not intersecting K, and X is aposyndetic if it is aposyndetic at each point with respect to every other point. A continuum X is said to be aposyndetic provided that for each point $p \in X$ and for each $q \in X \setminus \{p\}$ there exists a subcontinuum K of X and an open set U of X such that $p \in U \subset K \subset X \setminus \{q\}$ (see e.g.[65, Exercise 1.22, p. 12]).