Metrizable space: Difference between revisions
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===Symbol-free definition=== | ===Symbol-free definition=== | ||
A [[topological space]] is said to be '''metrizable''' if it occurs as the underlying topological space of a [[metric space]]. | A [[topological space]] is said to be '''metrizable''' if it occurs as the underlying topological space of a [[metric space]] via the [[metric induces topology|naturally induced topology]] with the basis open sets being the open balls with center in the space and finite positive radius. | ||
==Relation with other properties== | ==Relation with other properties== | ||
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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::completely metrizable space]] || arises from a [[complete metric space]] via the [[metric induces topology|naturally induced topology]] || [[completely metrizable implies metrizable]] || [[metrizable not implies completely metrizable]] || {{intermediate notions short|metrizable space|completely metrizable space}} | |||
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| [[Weaker than::sub-Euclidean space]] || homeomorphic to a subspace (via the [[subspace topology]]) of a [[Euclidean space]] <math>\R^n</math> || || || {{intermediate notions short|metrizable space|completely metrizable space}} | |||
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| [[Weaker than::closed sub-Euclidean space]] || homeomorphic to a [[closed subset]], under the [[subspace topology]], of a [[Euclidean space]] <math>\R^n</math> || || || {{intermediate notions short|metrizable space|closed sub-Euclidean space}} | |||
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| [[Weaker than::Manifold]] || || || || {{intermediate notions short|metrizable space|manifold}} | |||
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===Weaker properties=== | ===Weaker properties=== | ||
{| class="sortable" border="1" | |||
! Property !! Meaning!! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
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| [[Stronger than::ordered field-metrizable space]] || has a ''metric'' on it that takes nonnegative values in an ordered field || || || {{intermediate notions short|ordered field-metrizable space|metrizable space}} | |||
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| [[Stronger than::elastic space]] || || || || {{intermediate notions short|elastic space|metrizable space}} | |||
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| [[Stronger than::monotonically normal space]] || || [[metrizable implies monotonically normal]] || || {{intermediate notions short|monotonically normal space|metrizable space}} | |||
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| [[Stronger than::collectionwise normal space]] || any discrete collection of closed subsets can be separated by pairwise disjoint open subsets || (via monotonically normal) || || {{intermediate notions short|collectionwise normal space|metrizable space}} | |||
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| [[Stronger than::collectionwise Hausdorff space]] || the points in any discrete closed subset can be separated via pairwise disjoint open subsets || (via collectionwise normal) || || {{intermediate notions short|collectionwise Hausdorff space|metrizable space}} | |||
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| [[Stronger than::perfectly normal space]] || normal, and every closed subset is a [[G-delta subset]] || [[metrizable implies perfectly normal]] || || {{intermediate notions short|perfectly normal space|metrizable space}} | |||
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| [[Stronger than::perfect space]] || every singleton subset is a [[G-delta subset]] || (via perfectly normal) || || {{intermediate notions short|perfect space|metrizable space}} | |||
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| [[Stronger than::hereditarily normal space]] || every subspace is normal || [[metrizable implies hereditarily normal]] || || {{intermediate notions short|hereditarily normal space|metrizable space}} | |||
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| [[Stronger than::regular Lindelof space]] || [[regular space|regular]] and [[Lindelof space|Lindelof]] || || || {{intermediate notions short|regular Lindelof space|metrizable space}} | |||
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| [[Stronger than::paracompact Hausdorff space]] || [[paracompact space|paracompact]] and [[Hausdorff space|Hausdorff]] || || || {{intermediate notions short|paracompact Hausdorff space|metrizable space}} | |||
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| [[Stronger than::normal space]] || [[T1 space|T1]] and any two disjoint [[closed subset]]s can be separated using disjoint open subsets || [[metrizable implies normal]] || || {{intermediate notions short|normal space|metrizable space}} | |||
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| [[Stronger than::completely regular space]] || [[T1 space|T1]] and any point and closed subset not containing it can be separated via a continuous function to <math>[0,1]</math> || [[metrizable implies completely regular]] || || {{intermediate notions short|completely regular space|metrizable space}} | |||
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| [[Stronger than::regular space]] || [[T1 space|T1]] and any point and closed subset not containing it can be separated via disjoint open subsets || [[metrizable implies regular]] || || {{intermediate notions short|regular space|metrizable space}} | |||
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| [[Stronger than::Hausdorff space]] || any two distinct points can be separated via disjoint open subsets || [[metrizable implies Hausdorff]] || || {{intermediate notions short|Hausdorff space|metrizable space}} | |||
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| [[Stronger than::T1 space]] || every singleton subset is closed || || || {{intermediate notions short|T1 space|metrizable space}} | |||
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| [[Stronger than::Kolmogorov space]] || given any two points, there is an open subset containing one point and not the other || || || {{intermediate notions short|Kolmogorov space|metrizable space}} | |||
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| [[Stronger than::first-countable space]] || there is a countable basis, locally around each point || [[metrizable implies first-countable]] || || {{intermediate notions short|first-countable space|metrizable space}} | |||
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| [[Stronger than::submetrizable space]] || either itself metrizable, or metrizable upon passage to a coarser topology || || || {{intermediate notions short|submetrizable space|metrizable space}} | |||
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==Metaproperties== | ==Metaproperties== | ||
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{{subspace-closed}} | {{subspace-closed}} | ||
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. | 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. {{proofat|[[Topology from subspace metric equals subspace topology]]}} | ||
{{finite-DP-closed}} | {{finite-DP-closed}} | ||
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 <math>L^p</math>-norms (<math>1 \le p \le \infty</math>) to combine the individual metrics. | 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 <math>L^p</math>-norms (<math>1 \le p \le \infty</math>) to combine the individual metrics. {{proofat|[[Metrizability is finite-direct product-closed]]}} | ||
==References== | |||
===Textbook references=== | |||
* {{booklink-defined|Munkres}}, Page 120, Chapter 2, Section 20 (formal definition, along with [[metric space]]) | |||
Latest revision as of 23:49, 15 November 2015
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 via the naturally induced topology with the basis open sets being the open balls with center in the space and finite positive radius.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| completely metrizable space | arises from a complete metric space via the naturally induced topology | completely metrizable implies metrizable | metrizable not implies completely metrizable | |FULL LIST, MORE INFO |
| sub-Euclidean space | homeomorphic to a subspace (via the subspace topology) of a Euclidean space | |FULL LIST, MORE INFO | ||
| closed sub-Euclidean space | homeomorphic to a closed subset, under the subspace topology, of a Euclidean space | |FULL LIST, MORE INFO | ||
| Manifold | |FULL LIST, MORE INFO |
Weaker properties
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. For full proof, refer: Topology from subspace metric equals subspace topology
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. For full proof, refer: Metrizability is finite-direct product-closed
References
Textbook references
- Topology (2nd edition) by James R. MunkresMore info, Page 120, Chapter 2, Section 20 (formal definition, along with metric space)