Metrizable space: Difference between revisions
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===Stronger properties=== | ===Stronger properties=== | ||
* [[Completely metrizable space]] | * [[Weaker than::Completely metrizable space]] | ||
* [[sub-Euclidean space]] | * [[Weaker than::sub-Euclidean space]] | ||
* [[closed sub-Euclidean space]] | * [[Weaker than::closed sub-Euclidean space]] | ||
* [[Manifold]] | * [[Weaker than::Manifold]] | ||
===Weaker properties=== | ===Weaker properties=== | ||
* [[Ordered field-metrizable space]] | * [[Stronger than::Ordered field-metrizable space]] | ||
* [[Elastic space]] | * [[Stronger than::Elastic space]] | ||
* [[Monotonically normal space]]: {{proofat|[[Metrizable implies monotonically normal]]}} | * [[Stronger than::Monotonically normal space]]: {{proofat|[[Metrizable implies monotonically normal]]}} | ||
* [[Perfectly normal space]] | ** [[Stronger than::Collectionwise normal space]]: {{proofat|[[Metrizable implies collectionwise normal]]}} | ||
* [[Regular Lindelof space]] | ** [[Stronger than::Collectionwise Hausdorff space]]: {{proofat|[[Metrizable implies collectionwise Hausdorff]]}} | ||
* [[Paracompact Hausdorff space]] | * [[Stronger than::Perfectly normal space]] | ||
* [[Normal space]] | * [[Stronger than::Regular Lindelof space]] | ||
* [[Stronger than::Paracompact Hausdorff space]] | |||
* [[Stronger than::Normal space]] | |||
** [[Stronger than::Regular space]] | |||
** [[Stronger than::Hausdorff space]] | |||
==Metaproperties== | ==Metaproperties== | ||
Revision as of 15:09, 13 May 2009
This article defines a property of topological space that is pivotal (viz important) among currently studied properties of topological spaces
This article is about a basic definition in topology.
VIEW: Definitions built on this | Facts about this | Survey articles about this
View a complete list of basic definitions in topology
Definition
Symbol-free definition
A topological space is said to be metrizable if it occurs as the underlying topological space of a metric space.
Relation with other properties
Stronger properties
Weaker properties
- Ordered field-metrizable space
- Elastic space
- Monotonically normal space: For full proof, refer: Metrizable implies monotonically normal
- Collectionwise normal space: For full proof, refer: Metrizable implies collectionwise normal
- Collectionwise Hausdorff space: For full proof, refer: Metrizable implies collectionwise Hausdorff
- Perfectly normal space
- Regular Lindelof space
- Paracompact Hausdorff space
- Normal space
Metaproperties
Hereditariness
This property of topological spaces is hereditary, or subspace-closed. In other words, any subspace (subset with the subspace topology) of a topological space with this property also has this property.
View other subspace-hereditary properties of topological spaces
Any subspace of a metrizable space is metrizable. In fact, the subspace topology coincides with the topology induced from the metric obtained on the subset ,by restricting the metric from the whole space.
Products
This property of topological spaces is closed under taking finite products
A finite product of metrizable spaces is again metrizable. In fact, we can take the metric as, say, the sum of metric distances in each coordinate. More generally, we could use any of the -norms () to combine the individual metrics.
References
Textbook references
- Topology (2nd edition) by James R. MunkresMore info, Page 120, Chapter 2, Section 20 (formal definition, along with metric space)