Editing Initial topology (section)

===Initial uniform structure===

{{Main|Uniform space}}

If <math>\left(\mathcal{U}_i\right)_{i \in I}</math> is a family of [[uniform structure]]s on <math>X</math> indexed by <math>I \neq \varnothing,</math> then the {{em|[[least upper bound]] uniform structure}} of <math>\left(\mathcal{U}_i\right)_{i \in I}</math> is the coarsest uniform structure on <math>X</math> that is finer than each <math>\mathcal{U}_i.</math> This uniform always exists and it is equal to the [[Filter (set theory)|filter]] on <math>X \times X</math> generated by the [[filter subbase]] <math>{\textstyle \bigcup\limits_{i \in I} \mathcal{U}_i}.</math>{{sfn|Grothendieck|1973|p=3}} 
If <math>\tau_i</math> is the topology on <math>X</math> induced by the uniform structure <math>\mathcal{U}_i</math> then the topology on <math>X</math> associated with least upper bound uniform structure is equal to the least upper bound topology of <math>\left(\tau_i\right)_{i \in I}.</math>{{sfn|Grothendieck|1973|p=3}} 

Now suppose that <math>\left\{f_i : X \to Y_i\right\}</math> is a family of maps and for every <math>i \in I,</math> let <math>\mathcal{U}_i</math> be a uniform structure on <math>Y_i.</math> Then the {{em|initial uniform structure of the <math>Y_i</math> by the mappings <math>f_i</math>}} is the unique coarsest uniform structure <math>\mathcal{U}</math> on <math>X</math> making all <math>f_i : \left(X, \mathcal{U}\right) \to \left(Y_i, \mathcal{U}_i\right)</math> [[uniformly continuous]].{{sfn|Grothendieck|1973|p=3}} It is equal to the least upper bound uniform structure of the <math>I</math>-indexed family of uniform structures <math>f_i^{-1}\left(\mathcal{U}_i\right)</math> (for <math>i \in I</math>).{{sfn|Grothendieck|1973|p=3}} 
The topology on <math>X</math> induced by <math>\mathcal{U}</math> is the coarsest topology on <math>X</math> such that every <math>f_i : X \to Y_i</math> is continuous.{{sfn|Grothendieck|1973|p=3}} 
The initial uniform structure <math>\mathcal{U}</math> is also equal to the coarsest uniform structure such that the identity mappings <math>\operatorname{id} : \left(X, \mathcal{U}\right) \to \left(X, f_i^{-1}\left(\mathcal{U}_i\right)\right)</math> are uniformly continuous.{{sfn|Grothendieck|1973|p=3}} 

'''Hausdorffness''': The topology on <math>X</math> induced by the initial uniform structure <math>\mathcal{U}</math> is [[Hausdorff space|Hausdorff]] if and only if for whenever <math>x, y \in X</math> are distinct (<math>x \neq y</math>) then there exists some <math>i \in I</math> and some entourage <math>V_i \in \mathcal{U}_i</math> of <math>Y_i</math> such that <math>\left(f_i(x), f_i(y)\right) \not\in V_i.</math>{{sfn|Grothendieck|1973|p=3}} 
Furthermore, if for every index <math>i \in I,</math> the topology on <math>Y_i</math> induced by <math>\mathcal{U}_i</math> is Hausdorff then the topology on <math>X</math> induced by the initial uniform structure <math>\mathcal{U}</math> is Hausdorff if and only if the maps <math>\left\{f_i : X \to Y_i\right\}</math> [[#separate points|separate points]] on <math>X</math>{{sfn|Grothendieck|1973|p=3}} (or equivalently, if and only if the [[#evaluation map|evaluation map]] <math display=inline>f : X \to \prod_i Y_i</math> is injective)

'''Uniform continuity''': If <math>\mathcal{U}</math> is the initial uniform structure induced by the mappings <math>\left\{f_i : X \to Y_i\right\},</math> then a function <math>g</math> from some uniform space <math>Z</math> into <math>(X, \mathcal{U})</math> is [[uniformly continuous]] if and only if <math>f_i \circ g : Z \to Y_i</math> is uniformly continuous for each <math>i \in I.</math>{{sfn|Grothendieck|1973|p=3}} 

'''Cauchy filter''': A [[Filter (set theory)|filter]] <math>\mathcal{B}</math> on <math>X</math> is a [[Cauchy filter]] on <math>(X, \mathcal{U})</math> if and only if <math>f_i\left(\mathcal{B}\right)</math> is a Cauchy prefilter on <math>Y_i</math> for every <math>i \in I.</math>{{sfn|Grothendieck|1973|p=3}} 

'''Transitivity of the initial uniform structure''': If the word "topology" is replaced with "uniform structure" in the statement of "[[#Transitivity of the initial topology|transitivity of the initial topology]]" given above, then the resulting statement will also be true.