# Submetrizable space

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

### 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.