homeomorphism onto the subspace of the product space Separating points from closed sets If a space comes equipped with a topology, it is often useful to know whether or not the topology on is the initial topology induced by some family of maps on This section gives a sufficient (but not necessary) condition. A family of maps separates points from closed sets in if for all closed sets in and all there exists some such that where denotes the closure operator. Theorem. A family of continuous maps separates points from closed sets if and only if the cylinder sets for open in form a base for the topology on It follows that whenever separates points from closed sets, the space has the initial topology induced by the maps The converse fails, since generally the cylinder sets will only form a subbase (and not a base) for the initial topology. If the space is a T0 space, then any collection of maps that separates points from closed sets in must also separate points. In this case, the evaluation map will be an embedding. Initial uniform structure If is a family of uniform structures on indexed by then the of is the coarsest uniform structure on that is finer than each This uniform always exists and it is equal to the filter on generated by the filter subbase If is the topology on induced by the uniform structure then the topology on associated with least upper bound uniform structure is equal to the least upper bound topology of Now suppose that is a family of maps and for every let be a uniform structure on Then the is the unique coarsest uniform structure on making all uniformly continuous. It is equal to the least upper bound uniform structure of the -indexed family of uniform structures (for ). The topology on induced by is the coarsest topology on such that every is continuous. The initial uniform structure is also equal to the coarsest uniform structure such that the identity mappings are uniformly continuous. Hausdorffness: The topology on induced by the initial uniform structure is Hausdorff if and only if for whenever are distinct () then there exists some and some entourage of such that Furthermore, if for every index the topology on induced by is Hausdorff then the topology on induced by the initial uniform structure is Hausdorff if and only if the