In general topology and related areas of mathematics, the initial 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 which 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.
Read more about Initial Topology: Definition, Examples, Categorical Description
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