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An important concept in topology. A topological space is said to be metrizable if it is possible to define a metric on it which induces the original topology. Such spaces are particular nice topological spaces in many ways:

Theorem A metrizable space is completely normal, first countable and paracompact.

Theorem The following are equivalent for a metric space X:

  1. X is second countable
  2. X is Lindelof
  3. X is separable

Not all topological spaces are metrizable by a long way, particularly those stemming from algebra, number theory and set theory; the study of metrizable spaces (metric spaces) is mainly the department of analysis. It's important to know when a space can permit a metric, and much study has gone into metrization theorems giving conditions for a topological space to be metrizable. The most basic theorems are:

Theorem (Urysohn metrization) Let X be a regular, second countable topological space. Then X is metrizable.

Theorem (Nagata-Smirnov metrization) A topological space X is metrizable if and only if X is regular and has a basis which is countably locally finite (sigma-locally finite).

Theorem (Smirnov metrization) A topological space X is metrizable if and only if X is Hausdorff, paracompact, and locally metrizable.

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