Uniformly based space: Difference between revisions
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==Definition== | ==Definition== | ||
A [[topological space]] is said to be '''uniformly based''' if it has a basis of open sets, for which all basis elements are homeomorphic. | A [[topological space]] is said to be '''uniformly based''' if it has a basis of open sets, for which all basis elements are homeomorphic. The abstract space to which they are all homeomorphic is termed the '''basis space'''. | ||
==Relation with other properties== | ==Relation with other properties== | ||
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===Stronger properties=== | ===Stronger properties=== | ||
* [[Manifold]] | * [[Manifold]]: The basis space here is Euclidean space | ||
==Facts== | |||
For any property <math>p</math> of topological spaces, a uniformly based space satisfies the property ''locally'' <math>p</math> if and only if its basis space satisfies the property ''locally'' <math>p</math>. | |||
Revision as of 21:18, 27 October 2007
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 term is nonstandard and is being used locally within the wiki. For its use outside the wiki, please define the term when using it.
Definition
A topological space is said to be uniformly based if it has a basis of open sets, for which all basis elements are homeomorphic. The abstract space to which they are all homeomorphic is termed the basis space.
Relation with other properties
Stronger properties
- Manifold: The basis space here is Euclidean space
Facts
For any property of topological spaces, a uniformly based space satisfies the property locally if and only if its basis space satisfies the property locally .