Editing Initial topology (section)

===Separating points from closed sets===

If a space <math>X</math> comes equipped with a topology, it is often useful to know whether or not the topology on <math>X</math> is the initial topology induced by some family of maps on <math>X.</math> This section gives a sufficient (but not necessary) condition.

A family of maps <math>\left\{f_i : X \to Y_i\right\}</math> ''separates points from closed sets'' in <math>X</math> if for all [[closed set]]s <math>A</math> in <math>X</math> and all <math>x \not\in A,</math> there exists some <math>i</math> such that
<math display=block>f_i(x) \notin \operatorname{cl}(f_i(A))</math>
where <math>\operatorname{cl}</math> denotes the [[Closure (topology)|closure operator]].

:'''Theorem'''. A family of continuous maps <math>\left\{f_i : X \to Y_i\right\}</math> separates points from closed sets if and only if the cylinder sets <math>f_i^{-1}(V),</math> for <math>V</math> open in <math>Y_i,</math> form a [[Base (topology)|base for the topology]] on <math>X.</math>

It follows that whenever <math>\left\{f_i\right\}</math> separates points from closed sets, the space <math>X</math> has the initial topology induced by the maps <math>\left\{f_i\right\}.</math> The converse fails, since generally the cylinder sets will only form a subbase (and not a base) for the initial topology.

If the space <math>X</math> is a [[T0 space|T<sub>0</sub> space]], then any collection of maps <math>\left\{f_i\right\}</math> that separates points from closed sets in <math>X</math> must also separate points. In this case, the evaluation map will be an embedding.