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
Symbolfree 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
Weaker properties
Property 
Meaning 
Proof of implication 
Proof of strictness (reverse implication failure) 
Intermediate notions

ordered fieldmetrizable space 
has a metric on it that takes nonnegative values in an ordered field 


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elastic space 



Protometrizable spaceFULL LIST, MORE INFO

monotonically normal space 

metrizable implies monotonically normal 

Elastic space, Protometrizable spaceFULL 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 spaceFULL 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 spaceFULL LIST, MORE INFO

perfectly normal space 
normal, and every closed subset is a Gdelta subset 
metrizable implies perfectly normal 

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perfect space 
every singleton subset is a Gdelta subset 
(via perfectly normal) 

Perfectly normal spaceFULL 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 spaceFULL LIST, MORE INFO

regular Lindelof space 
regular and Lindelof 


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paracompact Hausdorff space 
paracompact and Hausdorff 


Elastic spaceFULL 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 spaceFULL LIST, MORE INFO

completely regular space 
T1 and any point and closed subset not containing it can be separated via a continuous function to 
metrizable implies completely regular 

Monotonically normal space, Normal Hausdorff space, Paracompact Hausdorff space, Tychonoff spaceFULL 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 spaceFULL 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 spaceFULL 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 spaceFULL 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 spaceFULL LIST, MORE INFO

firstcountable space 
there is a countable basis, locally around each point 
metrizable implies firstcountable 

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submetrizable space 
either itself metrizable, or metrizable upon passage to a coarser topology 


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Metaproperties
Hereditariness
This property of topological spaces is hereditary, or subspaceclosed. In other words, any subspace (subset with the subspace topology) of a topological space with this property also has this property.
View other subspacehereditary 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 norms () to combine the individual metrics. For full proof, refer: Metrizability is finitedirect productclosed
References
Textbook references
 Topology (2nd edition) by James R. Munkres^{More info}, Page 120, Chapter 2, Section 20 (formal definition, along with metric space)