Uniformly connected space

Type of uniform 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

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A compact uniform space is uniformly connected if and only if it is connected

## Examples

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* every connected space is uniformly connected
* the rational numbers and the irrational numbers are disconnected but uniformly connected

## See also

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* connectedness

## References

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

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