Initial topology
Coarsest topology making certain functions continuous / From Wikipedia, the free encyclopedia
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In general topology and related areas of mathematics, the initial topology (or induced topology[1][2] 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 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.