induced by the family . In particular: the quotient topology is the final topology on the quotient space induced by the quotient map. The final topology on a set induced by a family of -valued maps can be viewed as a far reaching generalization of the quotient topology, where multiple maps may be used instead of just one and where these maps are not required to be surjections. Given topological spaces , the disjoint union topology on the disjoint union is the final topology on the disjoint union induced by the natural injections. Given a family of topologies on a fixed set the final topology on with respect to the identity maps as ranges over call it is the infimum (or meet) of these topologies in the lattice of topologies on That is, the final topology is equal to the intersection The direct limit of any direct system of spaces and continuous maps is the set-theoretic direct limit together with the final topology determined by the canonical morphisms. Explicitly, this means that if is a direct system in the category Top of topological spaces and if is a direct limit of in the category Set of all sets, then by endowing with the final topology induced by becomes the direct limit of in the category Top. The étalé space of a sheaf is topologized by a final topology. A first-countable Hausdorff space is locally path-connected if and only if is equal to the final topology on induced by the set of all continuous maps where any such map is called a path in If a Hausdorff locally convex topological vector space is a Fréchet-Urysohn space then is equal to the final topology on induced by the set of all arcs in which by definition are continuous paths that are also topological embeddings. Properties Characterization via continuous maps Given functions from topological spaces to the set , the final topology on with respect to these functions satisfies the following property: a function from to some space is continuous if and only if is continuous for each This property characterizes the final topology in the sense that if a topology on satisfies the property above for all spaces and all functions , then the topology on is the final topology with respect to the Behavior under composition Suppose is a family of maps, and for every the topology on is the final