maps separate points on (or equivalently, if and only if the evaluation map is injective) Uniform continuity: If is the initial uniform structure induced by the mappings then a function from some uniform space into is uniformly continuous if and only if is uniformly continuous for each Cauchy filter: A filter on is a Cauchy filter on if and only if is a Cauchy prefilter on for every Transitivity of the initial uniform structure: If the word "topology" is replaced with "uniform structure" in the statement of "transitivity of the initial topology" given above, then the resulting statement will also be true. Categorical description In the language of category theory, the initial topology construction can be described as follows. Let be the functor from a discrete category to the category of topological spaces which maps . Let be the usual forgetful functor from to . The maps can then be thought of as a cone from to That is, is an object of —the category of cones to More precisely, this cone defines a -structured cosink in The forgetful functor induces a functor . The characteristic property of the initial topology is equivalent to the statement that there exists a universal morphism from to that is, a terminal object in the category Explicitly, this consists of an object in together with a morphism such that for any object in and morphism there exists a unique morphism such that the following diagram commutes: The assignment placing the initial topology on extends to a functor which is right adjoint to the forgetful functor In fact, is a right-inverse to ; since is the identity functor on See also References Bibliography External links General topology