Editing Initial topology (section)

===Characteristic property===

The initial topology on <math>X</math> can be characterized by the following characteristic property:<br>
A function <math>g</math> from some space <math>Z</math> to <math>X</math> is continuous if and only if <math>f_i \circ g</math> is continuous for each <math>i \in I.</math>{{sfn|Grothendieck|1973|p=2}} 

[[Image:InitialTopology-01.png|center|Characteristic property of the initial topology]]

Note that, despite looking quite similar, this is not a [[universal property]]. A categorical description is given below.

A [[Filter (set theory)|filter]] <math>\mathcal{B}</math> on <math>X</math> [[Convergent filter|converges to]] a point <math>x \in X</math> if and only if the [[prefilter]] <math>f_i(\mathcal{B})</math> [[Convergent prefilter|converges to]] <math>f_i(x)</math> for every <math>i \in I.</math>{{sfn|Grothendieck|1973|p=2}}