Editing Initial topology (section)

==Definition==

Given a set <math>X</math> and an [[indexed family]] <math>\left(Y_i\right)_{i \in I}</math> of [[topological space]]s with functions
<math display=block>f_i : X \to Y_i,</math>
the initial topology <math>\tau</math> on <math>X</math> is the [[coarsest topology]] on <math>X</math> such that each
<math display=block>f_i : (X, \tau) \to Y_i</math>
is [[continuous function (topology)|continuous]].

'''Definition in terms of open sets'''

If <math>\left(\tau_i\right)_{i \in I}</math> is a family of topologies <math>X</math> indexed by <math>I \neq \varnothing,</math> then the {{em|[[least upper bound]] topology}} of these topologies is the coarsest topology on <math>X</math> that is finer than each <math>\tau_i.</math> This topology always exists and it is equal to the [[subbase|topology generated by]] <math>{\textstyle \bigcap\limits_{i \in I} \tau_i}.</math>{{sfn|Grothendieck|1973|p=1}}

If for every <math>i \in I,</math> <math>\sigma_i</math> denotes the topology on <math>Y_i,</math> then <math>f_i^{-1}\left(\sigma_i\right) = \left\{f_i^{-1}(V) : V \in \sigma_i\right\}</math> is a topology on <math>X</math>, and the {{em|initial topology of the <math>Y_i</math> by the mappings <math>f_i</math>}} is the least upper bound topology of the <math>I</math>-indexed family of topologies <math>f_i^{-1}\left(\sigma_i\right)</math> (for <math>i \in I</math>).{{sfn|Grothendieck|1973|p=1}} 
Explicitly, the initial topology is the collection of open sets [[subbase|generated]] by all sets of the form <math>f_i^{-1}(U),</math> where <math>U</math> is an [[open set]] in <math>Y_i</math> for some <math>i \in I,</math> under finite intersections and arbitrary unions. 

Sets of the form <math>f_i^{-1}(V)</math> are often called {{em|[[cylinder set]]s}}. If <math>I</math> contains [[Singleton set|exactly one element]], then all the open sets of the initial topology <math>(X, \tau)</math> are cylinder sets.