Initial topology

In general topology and related areas of mathematics, the initial topology (or weak topology or limit topology or projective topology) on a set X {displaystyle X} , with respect to a family of functions on X {displaystyle X} , is the coarsest topology on X that makes those functions continuous. In general topology and related areas of mathematics, the initial topology (or weak topology or limit topology or projective topology) on a set X {displaystyle X} , with respect to a family of functions on X {displaystyle X} , 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 X {displaystyle X} is the finest topology on X {displaystyle X} that makes those functions continuous. Given a set X and an indexed family (Yi)i∈I of topological spaces with functions the initial topology τ {displaystyle au } on X {displaystyle X} is the coarsest topology on X such that each is continuous. Explicitly, the initial topology is the collection of open sets generated by all sets of the form f i − 1 ( U ) {displaystyle f_{i}^{-1}(U)} , where U {displaystyle U} is an open set in Y i {displaystyle Y_{i}} for some i ∈ I, under finite intersections and arbitrary unions. The sets f i − 1 ( U ) {displaystyle f_{i}^{-1}(U)} are often called cylinder sets.If I contains exactly one element, all the open sets of ( X , τ ) {displaystyle (X, au )} are cylinder sets.