Unicoherent space

In mathematics, a unicoherent space is a topological space $X$ that is connected and in which 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.

If a unicoherent space is more strongly hereditarily unicoherent (meaning that every subcontinuum is unicoherent) and arcwise connected, then it is called a dendroid. If in addition it is locally connected then it is called a dendrite. The Phragmen–Brouwer theorem states that, for locally connected spaces, unicoherence is equivalent to a separation property of the closed sets of the space.

References

Insall, Matt. "Unicoherent Space". MathWorld.

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