Submetrizable space: Difference between revisions
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==Definition== | ==Definition== | ||
A [[topological space]] is said to be '''submetrizable''' if it is either a [[metrizable space]] to begin with or we can choose a [[coarser topology]] on the space and thus make it a [[metrizable space]]. | |||
A [[topological space]] is said to be '''submetrizable''' if we can choose a [[coarser topology]] on the space and thus make it a [[metrizable space]]. | |||
==Relation with other properties== | ==Relation with other properties== | ||
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===Stronger properties=== | ===Stronger properties=== | ||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
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| [[Weaker than::metrizable space]] || underlying topology of a [[metric space]] || || || {{intermediate notions short|submetrizable space|metrizable space}} | |||
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| [[Weaker than::manifold]] || (via metrizable) || || || {{intermediate notions short|submetrizable space|manifold}} | |||
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===Weaker properties=== | |||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
|- | |||
| [[Stronger than::Urysohn space]] || distinct points can be separated by a continuous function to the reals || || || {{intermediate notions short|Urysohn space|submetrizable space}} | |||
|- | |||
| [[Stronger than::collectionwise Hausdorff space]] || discrete closed subset can be separated by pairwise disjoint open subsets || || || {{intermediate notions short|collectionwise Hausdorff space|submetrizable space}} | |||
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| [[Stronger than::Hausdorff space]] || distinct points can be separated by disjoint open subsets || || || {{intermediate notions short|Hausdorff space|submetrizable space}} | |||
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| [[Stronger than::T1 space]] || points are closed || || || {{intermediate notions short|T1 space|submetrizable space}} | |||
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==Metaproperties== | ==Metaproperties== | ||
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This follows from the fact that a direct product of metrizable spaces is metrizable. | This follows from the fact that a direct product of metrizable spaces is metrizable. | ||
{{subspace-closed}} | |||
This follows from the fact that any subspace of a metrizable space is metrizable. | |||
{{refining-preserved}} | {{refining-preserved}} | ||
This follows immediately from the definition. | This follows immediately from the definition. | ||
Latest revision as of 02:42, 25 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 a variation of metrizability. View other variations of metrizability
Definition
A topological space is said to be submetrizable if it is either a metrizable space to begin with or we can choose a coarser topology on the space and thus make it a metrizable space.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| metrizable space | underlying topology of a metric space | |FULL LIST, MORE INFO | ||
| manifold | (via metrizable) | Metrizable space|FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Urysohn space | distinct points can be separated by a continuous function to the reals | Functionally Hausdorff space|FULL LIST, MORE INFO | ||
| collectionwise Hausdorff space | discrete closed subset can be separated by pairwise disjoint open subsets | |FULL LIST, MORE INFO | ||
| Hausdorff space | distinct points can be separated by disjoint open subsets | Functionally Hausdorff space|FULL LIST, MORE INFO | ||
| T1 space | points are closed | Functionally Hausdorff space, Hausdorff space|FULL LIST, MORE INFO |
Metaproperties
Products
This property of topological spaces is closed under taking arbitrary products
View all properties of topological spaces closed under products
This follows from the fact that a direct product of metrizable spaces is metrizable.
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
This follows from the fact that any subspace of a metrizable space is metrizable.
Refining
This property of topological spaces is preserved under refining, viz, if a set with a given topology has the property, the same set with a finer topology also has the property
View all refining-preserved properties of topological spaces OR View all coarsening-preserved properties of topological spaces
This follows immediately from the definition.