Unicoherent Space

Let  be a connected topological space. Then  is unicoherent provided that for any closed connected subsets  and  of , if , then  is connected.

An interval, say [0,1], is unicoherent, but a circle, say , is not unicoherent. An interesting example of a unicoherent space is a ray winding down on a circle. Specifically, let , where . Then the space , illustrated above, is unicoherent.

REFERENCES:

Charatonik, J. J. and Prajs, J. R. "On Local Connectedness of Absolute Retracts." Pacific J. Math. 201, 83-88, 2001.

Mackowiak, T. "Retracts of Hereditarily Unicoherent Continua." Bull. Acad. Polon. Sci. Ser. Sci. Math. 28, 177-183, 1980.