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 ${\displaystyle X}$, with respect to a family of functions on ${\displaystyle X}$, 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 construction is called the final topology.

Given a set X and an indexed family (Y_i)_i∈I of topological spaces with functions

the initial topology τ on ${\displaystyle X}$ is the coarsest topology on X such that each

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