metrizable

Metrizable describes a property of a topological space. A space X is metrizable if there exists a metric d on X such that the topology on X coincides with the one generated by the open balls of d. In other words, X is homeomorphic to a metric space.

If X is metrizable, then it is regular and T1, and many familiar spaces are metrizable. Examples include Euclidean spaces R^n, normed and metric spaces in analysis, and every discrete space. On the other hand, not all topological spaces are metrizable; common non-examples include the long line and, in general, uncountable products of uncountably many spaces with the product topology.

Two key metrization theorems provide practical criteria. The Urysohn Metrization Theorem states that any regular space with a countable base (second countable) is metrizable. The Nagata–Smirnov Metrization Theorem offers a broader criterion: a space is metrizable if and only if it is regular and possesses a σ-discrete (alternatively, σ-locally finite) base. These results give ways to recognize metrizability from topological structure rather than constructing a metric directly.

Related concepts include separability and second countability, which interact with metrizability in important ways. For instance, every separable metrizable space is second countable, but a metrizable space need not be separable or second countable in general.

In summary, metrizability links abstract topology with metric geometry, enabling the use of distances, convergence, and completeness to study topological properties.