Totally disconnected space: Difference between revisions
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===Stronger properties=== | ===Stronger properties=== | ||
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
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| [[Weaker than::discrete space]] || || [[discrete implies totally disconnected]] || [[totally disconnected not implies discrete]] || | |||
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===Weaker properties=== | |||
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
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| [[Stronger than::T1 space]] || all points are closed || [[totally disconnected implies T1]] || [[T1 not implies totally disconnected]] || | |||
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==Examples== | ==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. | 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=== | |||
* {{booklink|Munkres}}, Page 152, Exercise 5 (definition introduced in exercise) | |||
Latest revision as of 20:02, 13 January 2012
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)