In general topology and related areas of mathematics, the initial topology (or induced topology or weak topology or limit topology or projective topology) on a set with respect to a family of functions on is the coarsest topology on X that makes those functions continuous. The subspace topology and product topology constructions are both special cases of initial topologies. Indeed, the initial topology construction can be viewed as a generalization of these. The dual notion is the final topology, which for a given family of functions mapping to a set is the finest topology on that makes those functions continuous. Definition Given a set and an indexed family of topological spaces with functions the initial topology on is the coarsest topology on such that each is continuous. Definition in terms of open sets If is a family of topologies indexed by then the of these topologies is the coarsest topology on that is finer than each This topology always exists and it is equal to the topology generated by If for every denotes the topology then is a topology on and the is the least upper bound topology of the -indexed family of topologies (for ). Explicitly, the initial topology is the collection of open sets generated by all sets of the form where is an open set in for some under finite intersections and arbitrary unions. Sets of the form are often called . If contains exactly one element, then all the open sets of the initial topology are cylinder sets. Examples Several topological constructions can be regarded as special cases of the initial topology. The subspace topology is the initial topology on the subspace with respect to the inclusion map. The product topology is the initial topology with respect to the family of projection maps. The inverse limit of any inverse system of spaces and continuous maps is the set-theoretic inverse limit together with the initial topology determined by the canonical morphisms. The weak topology on a locally convex space is the initial topology with respect to the continuous linear forms of its dual space. Given a family of topologies on a fixed set the initial topology on with respect to the functions is the supremum (or join) of the topologies in the lattice of topologies on That is, the initial topology is the topology generated by the union of the topologies A topological space is completely regular