title: Unicoherent space
text: # 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.
  
  