Editing Initial topology (section)

{{Short description|Coarsest topology making certain functions continuous}}
In [[general topology]] and related areas of [[mathematics]], the '''initial topology''' (or '''induced topology'''<ref name=Rudin>{{Rudin Walter Functional Analysis| at=sections 3.8 and 3.11}}</ref><ref name=ad>{{cite book |chapter-url=https://link.springer.com/chapter/10.1007%2F978-0-8176-8126-5_3 |last=Adamson |first=Iain T.  |title=A General Topology Workbook |chapter=Induced and Coinduced Topologies |date=1996 |publisher=Birkhäuser, Boston, MA |access-date=July 21, 2020 |quote=...&nbsp;the topology induced on E by the family of mappings&nbsp;... |doi=10.1007/978-0-8176-8126-5_3|pages=23–30 |isbn=978-0-8176-3844-3 }}</ref> or '''strong topology''' or '''limit topology''' or '''projective topology''') on a [[Set (mathematics)|set]] <math>X,</math> with respect to a family of functions on <math>X,</math> is the [[coarsest topology]] on <math>X</math> that makes those functions [[Continuous function (topology)|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 [[Duality (mathematics)|dual]] notion is the [[final topology]], which for a given family of functions mapping to a set <math>Y</math> is the [[finest topology]] on <math>Y</math> that makes those functions continuous.