Totally disconnected space
From Topospaces
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
Contents
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. Munkres^{More info}, Page 152, Exercise 5 (definition introduced in exercise)