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:
- X is second countable
- X is Lindelof
- 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.