Submetrizable space

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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.