Editing Initial topology (section)

===Transitivity of the initial topology===

If <math>X</math> has the initial topology induced by the <math>I</math>-indexed family of mappings <math>\left\{f_i : X \to Y_i\right\}</math> and if for every <math>i \in I,</math> the topology on <math>Y_i</math> is the initial topology induced by some <math>J_i</math>-indexed family of mappings <math>\left\{g_j : Y_i \to Z_j\right\}</math> (as <math>j</math> ranges over <math>J_i</math>), then the initial topology on <math>X</math> induced by <math>\left\{f_i : X \to Y_i\right\}</math> is equal to the initial topology induced by the <math>{\textstyle \bigcup\limits_{i \in I} J_i}</math>-indexed family of mappings <math>\left\{g_j \circ f_i : X \to Z_j\right\}</math> as <math>i</math> ranges over <math>I</math> and <math>j</math> ranges over <math>J_i.</math>{{sfn|Grothendieck|1973|pp=1-2}} 
Several important corollaries of this fact are now given. 

In particular, if <math>S \subseteq X</math> then the subspace topology that <math>S</math> inherits from <math>X</math> is equal to the initial topology induced by the [[inclusion map]] <math>S \to X</math> (defined by <math>s \mapsto s</math>). Consequently, if <math>X</math> has the initial topology induced by <math>\left\{f_i : X \to Y_i\right\}</math> then the subspace topology that <math>S</math> inherits from <math>X</math> is equal to the initial topology induced on <math>S</math> by the restrictions <math>\left\{\left.f_i\right|_S : S \to Y_i\right\}</math> of the <math>f_i</math> to <math>S.</math>{{sfn|Grothendieck|1973|p=2}} 

The [[product topology]] on <math>\prod_i Y_i</math> is equal to the initial topology induced by the canonical projections <math>\operatorname{pr}_i : \left(x_k\right)_{k \in I} \mapsto x_i</math> as <math>i</math> ranges over <math>I.</math>{{sfn|Grothendieck|1973|p=2}}
Consequently, the initial topology on <math>X</math> induced by <math>\left\{f_i : X \to Y_i\right\}</math> is equal to the inverse image of the product topology on <math>\prod_i Y_i</math> by the [[#evaluation map|evaluation map]] <math display=inline>f : X \to \prod_i Y_i\,.</math>{{sfn|Grothendieck|1973|p=2}}  Furthermore, if the maps <math>\left\{f_i\right\}_{i \in I}</math> [[#separate points|separate points]] on <math>X</math> then the evaluation map is a [[homeomorphism]] onto the subspace <math>f(X)</math> of the product space <math>\prod_i Y_i.</math>{{sfn|Grothendieck|1973|p=2}}