In mathematics, the initial topology on a set is the weakest topology that makes one or more specified functions on that set continuous.

Let be a family of functions, each from a fixed set X to a topological space Y_i. The initial topology on X (with respect to the family of functions) is the weakest topology such that every f_i is continuous. The topology is generated by sets of the form , where U is an open set in Y_i.

Several topological constructions can be regarded as special cases of the initial topology. For example, the subspace topology for is the initial topology with respect to the inclusion map. The product topology is the initial topology with respect to the family of projection maps.

See also

• final topology