Metrizable space

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.
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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 ${\mathbb{R}}^n$			|FULL LIST, MORE INFO
closed sub-Euclidean space	homeomorphic to a closed subset, under the subspace topology, of a Euclidean space ${\mathbb{R}}^n$			|FULL LIST, MORE INFO
Manifold				|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
ordered field-metrizable space	has a metric on it that takes nonnegative values in an ordered field			|FULL LIST, MORE INFO
elastic space				Protometrizable space|FULL LIST, MORE INFO
monotonically normal space		metrizable implies monotonically normal		Elastic space, Protometrizable space|FULL LIST, MORE INFO
collectionwise normal space	any discrete collection of closed subsets can be separated by pairwise disjoint open subsets	(via monotonically normal)		Elastic space, Hereditarily collectionwise normal space, Monotonically normal space, Paracompact Hausdorff space, Protometrizable space|FULL LIST, MORE INFO
collectionwise Hausdorff space	the points in any discrete closed subset can be separated via pairwise disjoint open subsets	(via collectionwise normal)		Collectionwise normal space, Hereditarily collectionwise normal space, Monotonically normal space, Submetrizable space|FULL LIST, MORE INFO
perfectly normal space	normal, and every closed subset is a G-delta subset	metrizable implies perfectly normal		|FULL LIST, MORE INFO
perfect space	every singleton subset is a G-delta subset	(via perfectly normal)		Perfectly normal space|FULL LIST, MORE INFO
hereditarily normal space	every subspace is normal	metrizable implies hereditarily normal		Elastic space, Hereditarily collectionwise normal space, Monotonically normal space, Perfectly normal space, Protometrizable space|FULL LIST, MORE INFO
regular Lindelof space	regular and Lindelof			|FULL LIST, MORE INFO
paracompact Hausdorff space	paracompact and Hausdorff			Elastic space|FULL LIST, MORE INFO
normal space	T1 and any two disjoint closed subsets can be separated using disjoint open subsets	metrizable implies normal		Collectionwise normal space, Elastic space, Hereditarily collectionwise normal space, Hereditarily normal space, Monotonically normal space, Paracompact Hausdorff space, Perfectly normal space, Protometrizable space|FULL LIST, MORE INFO
completely regular space	T1 and any point and closed subset not containing it can be separated via a continuous function to $[0,1]$	metrizable implies completely regular		Monotonically normal space, Normal Hausdorff space, Paracompact Hausdorff space, Tychonoff space|FULL LIST, MORE INFO
regular space	T1 and any point and closed subset not containing it can be separated via disjoint open subsets	metrizable implies regular		Completely regular space, Monotonically normal space, Moore space, Normal Hausdorff space, Regular Hausdorff space, Tychonoff space|FULL LIST, MORE INFO
Hausdorff space	any two distinct points can be separated via disjoint open subsets	metrizable implies Hausdorff		Collectionwise Hausdorff space, Completely regular space, Functionally Hausdorff space, Monotonically normal space, Moore space, Normal Hausdorff space, Regular Hausdorff space, Submetrizable space, Tychonoff space|FULL LIST, MORE INFO
T1 space	every singleton subset is closed			Completely regular space, Functionally Hausdorff space, Hausdorff space, Normal Hausdorff space, Regular Hausdorff space, Submetrizable space|FULL LIST, MORE INFO
Kolmogorov space	given any two points, there is an open subset containing one point and not the other			Functionally Hausdorff space, Hausdorff space, Normal Hausdorff space, Regular Hausdorff space|FULL LIST, MORE INFO
first-countable space	there is a countable basis, locally around each point	metrizable implies first-countable		|FULL LIST, MORE INFO
submetrizable space	either itself metrizable, or metrizable upon passage to a coarser topology			|FULL LIST, MORE INFO

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 $L^p$-norms ($1\le p\le \mathrm{\infty }$) to combine the individual metrics. For full proof, refer: Metrizability is finite-direct product-closed

References

Textbook references

• Topology (2nd edition) by James R. Munkres^{More info}, Page 120, Chapter 2, Section 20 (formal definition, along with metric space)