A theorem named for this property is proven by invoking a lemma which states that normality is equivalent to disjoint closed subsets being separable by continuous functions.
A space that has this property locally will have it globally if it is paracompact and Hausdorff. The weak topology on a normed infinite dimensional space does not possess this property. Having a countably, locally finite basis is a necessary condition for having this property by the Nagata-Smirnov theorem. A theorem named for this property is proven by invoking a lemma which states that normality is equivalent to disjoint closed subsets being separable by continuous functions. That lemma is named for Pavel Urysohn and is used to show that a space being regular and countable is a sufficient criterion to have this property, a theorem that is also named for Pavel Urysohn. A space has this property, if, for any pair of points, the namesake function doesn't violate the triangle inequality, is symmetric, and is non-negative. For 10 points, name this property of spaces that allows one to define a function that returns a distance value.
A space that has this property locally will have it globally if it is paracompact and Hausdorff.
metrizable [or word forms; or the space has a metric; or the space is homeomorphic to a metric space]