Editing Initial topology (section)

==Examples==

Several topological constructions can be regarded as special cases of the initial topology.
* The [[subspace topology]] is the initial topology on the subspace with respect to the [[inclusion map]].
* The [[product topology]] is the initial topology with respect to the family of [[projection map]]s.
* The [[inverse limit]] of any [[inverse system]] of spaces and continuous maps is the set-theoretic inverse limit together with the initial topology determined by the canonical morphisms.
* The [[weak topology]] on a [[locally convex space]] is the initial topology with respect to the [[continuous linear form]]s of its [[dual space]].
* Given a [[indexed family|family]] of topologies <math>\left\{\tau_i\right\}</math> on a fixed set <math>X</math> the initial topology on <math>X</math> with respect to the functions <math>\operatorname{id}_i : X \to \left(X, \tau_i\right)</math> is the [[supremum]] (or join) of the topologies <math>\left\{\tau_i\right\}</math> in the [[lattice of topologies]] on <math>X.</math> That is, the initial topology <math>\tau</math> is the topology generated by the [[union (set theory)|union]] of the topologies <math>\left\{\tau_i\right\}.</math>
* A topological space is [[completely regular]] if and only if it has the initial topology with respect to its family of ([[bounded function|bounded]]) real-valued continuous functions.
* Every topological space <math>X</math> has the initial topology with respect to the family of continuous functions from <math>X</math> to the [[Sierpiński space]].