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.

Relation with other properties

Stronger properties

• Completely metrizable space
• sub-Euclidean space
• closed sub-Euclidean space
• Manifold

Weaker properties

• Ordered field-metrizable space
• Elastic space
• Monotonically normal space: For full proof, refer: Metrizable implies monotonically normal
  • Collectionwise normal space: For full proof, refer: Metrizable implies collectionwise normal
  • Collectionwise Hausdorff space: For full proof, refer: Metrizable implies collectionwise Hausdorff
• Perfectly normal space
  • Perfect space
  • Hereditarily normal space
• Regular Lindelof space
• Paracompact Hausdorff space
• Normal space
  • Completely regular space
  • Regular space
  • Hausdorff space
  • T1 space
  • Kolmogorov space
• First-countable space: For full proof, refer: Metrizable implies first-countable

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)