Every Tychonoff space of weight $\leq m$ can embeding to $I^m$

Question

How to prove the second paragraph question, that is, why $(\mathscr U^m)_X$ induces the original topology on $X$?As we still don't know $X\subseteq I^m$ is correct or not?

Answer 1

The embedding follows from Thm. 2.2.23 in Engelking (which I see you are quoting here). Here it is just remarked that this embedding is also valid as uniform spaces as well as topological spaces.