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 | ||
| 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.