Uniformly connected space

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