A sufficient criterion for a topological space to admit a compatible metric: a second-countable, regular Hausdorff space (equivalently regular T1 with a countable base) is metrizable — there exists a metric that induces the given topology.

Principle

A countable base allows one to index and combine local separation data into a single metric; regularity (separation of points and closed sets by neighborhoods) ensures the local control needed to make the constructed distance consistent with the topology.

Demonstration

Any finite-dimensional manifold or R^n is second-countable and regular Hausdorff, so the theorem guarantees the existence of a metric compatible with its standard topology; more structurally, one can build a metric by summing weighted Urysohn-style functions associated to a countable base.

Misapplication

Assuming metrizability follows from weaker hypotheses such as first-countability alone, or using the theorem for spaces lacking a countable base or regularity; confusing second-countability with separability (they are related but not identical).

Consequence

When applicable, the theorem lets one import metric-space results (sequences, completeness, Baire category arguments) to the topological setting and simplifies analysis by enabling concrete distance-based constructions and invariants.

Reversal

The converse fails in general: metrizable spaces need not be second-countable in some set-theoretic contexts, and metrizability can be characterized by other criteria (Nagata–Smirnov, Bing) that trade off countability for other covering properties.

Boundary

Provides sufficient but not universally necessary conditions; it requires a countable base and regularity/Hausdorff hypotheses and does not cover other metrization criteria or pathological examples where different axioms produce metrizability.

Semantic Tension

Competes with other metrization theorems (Nagata–Smirnov, Bing) that use different hypotheses (e.g., development, paracompactness) — one must choose the appropriate criterion depending on available structural hypotheses.

Synthesis

Urysohn's Metrization Theorem asserts that having a countable base together with regular separation properties suffices to produce a metric inducing the topology, turning certain abstract topological spaces into metric spaces and enabling metric techniques.