Totally disconnected space

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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)