Unicoherent

A topological space $X$ is said to be unicoherent if it is connected and the following property holds:

For any closed, connected $A, B \subset X$ with $X=A \cup B$ , the intersection $A \cap B$ is connected.

For example, any closed interval on the real line is unicoherent, but a circle is not.

Eric W. Weisstein et al. "Unicoherent Space." From MathWorld--A Wolfram Web Resource. [1]

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