Hereditarily Unicoherent Continuum

Let  be a continuum (i.e., a compact connected metric space). Then  is hereditarily unicoherent provided that every subcontinuum of  is unicoherent.

Any hereditarily unicoherent continuum is a unicoherent space, but there are unicoherent continua that are not hereditarily unicoherent. For example, the unit interval is hereditarily unicoherent, but a ray winding down on a circle is not hereditarily unicoherent, even though it is unicoherent. (This is due to the fact that a circle is not unicoherent.)