In [Tka] the author writes:

"Every topological space $X$ can be represented as an open continuous image of a completely regular submetrizable space $Y$ (in other words, $Y$ admits a continuous one-to-one mapping onto a metrizable space) — the corresponding construction is given on p. 331 of [Eng]".

But I can not find this statement on this page. Is there a source in which this statement is explicitly formulated and proved?

[Tka] M. G. Tkachenko. Topological groups for topologists: part I, Bol. Soc. Mat. Mexicana (3), 5, 1999, 237-279.

[Eng] R. Engelking, General Topology, Heldermann Verlag, Berlin 1989.