Editing Initial topology (section)

===Evaluation===

By the universal property of the [[product topology]], we know that any family of continuous maps <math>f_i : X \to Y_i</math> determines a unique continuous map
<math display=block>\begin{alignat}{4}
f :\;&& X &&\;\to    \;& \prod_i Y_i \\[0.3ex]
     && x &&\;\mapsto\;& \left(f_i(x)\right)_{i \in I} \\
\end{alignat}</math>

This map is known as the '''{{visible anchor|evaluation map}}'''.{{cn|reason=Such a counterintuitive term must be reliably sourced|date=February 2024}}

A family of maps <math>\{f_i : X \to Y_i\}</math> is said to ''[[Separating set|{{visible anchor|separate points}}]]'' in <math>X</math> if for all <math>x \neq y</math> in <math>X</math> there exists some <math>i</math> such that <math>f_i(x) \neq f_i(y).</math> The family <math>\{f_i\}</math> separates points if and only if the associated evaluation map <math>f</math> is [[injective]].

The evaluation map <math>f</math> will be a [[topological embedding]] if and only if <math>X</math> has the initial topology determined by the maps <math>\{f_i\}</math> and this family of maps separates points in <math>X.</math>