unicoherent

A connected topological space S is said to be unicoherent provided that whenever A and B are closed, connected subsets of S such that $S=A\cup B$, then $A\cap B$ is connected. Let a continuum X and its subcontinuum Y be given. Then X is said to be unicoherent at Y provided that for each pair of proper subcontinua A and B of X such that $X = A \cup B$ the intersection $A \cap B \cap Y$ is connected (see [72, p. 146]).