# Totally disconnected space

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

This is an opposite of connectedness

## Definition

A topological space is said to be totally disconnected if its connected components are one-point sets.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
discrete space discrete implies totally disconnected totally disconnected not implies discrete

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
T1 space all points are closed totally disconnected implies T1 T1 not implies totally disconnected

## Examples

The rational numbers form a totally disconnected space. In fact, any irrational number gives a disconnection by partitioning the rational numbers into two open subsets -- the subset of numbers less than the given irrational and the subset of numbers greater than the given irrational.

## References

### Textbook references

• Topology (2nd edition) by James R. MunkresMore info, Page 152, Exercise 5 (definition introduced in exercise)