| dbo:abstract
|
- In mathematics, the partition topology is a topology that can be induced on any set X by partitioning X into disjoint subsets P; these subsets form the basis for the topology. There are two important examples which have their own names:
* The odd–even topology is the topology where and Equivalently, .
* The deleted integer topology is defined by letting and . The trivial partitions yield the discrete topology (each point of X is a set in P, so ) or indiscrete topology (the entire set X is in P, so ). Any set X with a partition topology generated by a partition P can be viewed as a pseudometric space with a pseudometric given by: This is not a metric unless P yields the discrete topology. The partition topology provides an important example of the independence of various separation axioms. Unless P is trivial, at least one set in P contains more than one point, and the elements of this set are topologically indistinguishable: the topology does not separate points. Hence X is not a Kolmogorov space, nor a T1 space, a Hausdorff space or an Urysohn space. In a partition topology the complement of every open set is also open, and therefore a set is open if and only if it is closed. Therefore, X is regular, completely regular, normal and completely normal. X/P is the discrete topology. (en)
|
| dbo:wikiPageID
| |
| dbo:wikiPageLength
|
- 2787 (xsd:nonNegativeInteger)
|
| dbo:wikiPageRevisionID
| |
| dbo:wikiPageWikiLink
| |
| dbp:wikiPageUsesTemplate
| |
| dcterms:subject
| |
| rdf:type
| |
| rdfs:comment
|
- In mathematics, the partition topology is a topology that can be induced on any set X by partitioning X into disjoint subsets P; these subsets form the basis for the topology. There are two important examples which have their own names:
* The odd–even topology is the topology where and Equivalently, .
* The deleted integer topology is defined by letting and . The trivial partitions yield the discrete topology (each point of X is a set in P, so ) or indiscrete topology (the entire set X is in P, so ). This is not a metric unless P yields the discrete topology. (en)
|
| rdfs:label
| |
| owl:sameAs
| |
| prov:wasDerivedFrom
| |
| foaf:isPrimaryTopicOf
| |
| is dbo:wikiPageRedirects
of | |
| is dbo:wikiPageWikiLink
of | |
| is foaf:primaryTopic
of | |
