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

See also

• connectedness

References

1. Cantor, Georg Über Unendliche, lineare punktmannigfaltigkeiten, Mathematische Annalen. 21 (1883) 545-591.