if and only if it has the initial topology with respect to its family of (bounded) real-valued continuous functions. Every topological space has the initial topology with respect to the family of continuous functions from to the Sierpiński space. Properties Characteristic property The initial topology on can be characterized by the following characteristic property: A function from some space to is continuous if and only if is continuous for each Note that, despite looking quite similar, this is not a universal property. A categorical description is given below. A filter on converges to a point if and only if the prefilter converges to for every Evaluation By the universal property of the product topology, we know that any family of continuous maps determines a unique continuous map This map is known as the . A family of maps is said to in if for all in there exists some such that The family separates points if and only if the associated evaluation map is injective. The evaluation map will be a topological embedding if and only if has the initial topology determined by the maps and this family of maps separates points in Hausdorffness If has the initial topology induced by and if every is Hausdorff, then is a Hausdorff space if and only if these maps separate points on Transitivity of the initial topology If has the initial topology induced by the -indexed family of mappings and if for ever the topology on is the initial topology induced by some -indexed family of mappings (as ranges over ), then the initial topology on induced by is equal to the initial topology induced by the -indexed family of mappings as ranges over and ranges over Several important corollaries of this fact are now given. In particular, if then the subspace topology that inherits from is equal to the initial topology induced by the inclusion map (defined by ). Consequently, if has the initial topology induced by then the subspace topology that inherits from is equal to the initial topology induced on by the restrictions of the to The product topology on is equal to the initial topology induced by the canonical projections as ranges over Consequently, the initial topology on induced by is equal to the inverse image of the product topology on by the evaluation map Furthermore, if the maps separate points on then the evaluation map is a