In topology and related areas of mathematics, a uniformly connected space or Cantor connected space is a uniform space U such that every uniformly continuous function from U to a discrete uniform space is constant. A uniform space U is called uniformly disconnected if it is not uniformly connected.
Properties A compact uniform space is uniformly connected if and only if it is connected.
Examples Every connected space is uniformly connected. The rational numbers and the irrational numbers are disconnected but uniformly connected.
See also Connectedness
References Cantor, Georg Über Unendliche, lineare punktmannigfaltigkeiten, Mathematische Annalen. 21 (1883) 545-591.