Open images of submetrizable spaces

Question

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.

Answer 1

It seems quite likely that it was intended to be page 331 of the Engelking's book published in 1977 in PWN, Warszava¹ - rather then the later edition by Heldermann (revised and completed edition, 1989).

I will quote in full exercise 4.2.D (page 331 in the older edition, page 264 in the newer edition):

4.2.D. (a) (Ponomarev [1960]) Prove that a $T_0$-space $X$ is first countable if and only if $X$ is a continuous image of a metrizable space under an open mapping. Hint. Let $X$ be a first-countable space and let $\{U_s\}_{s\in S}$ be a base for $X$. Consider the Baire space $B(\mathfrak m) = \prod_{i=1}^\infty X_i$ where $X_i = S$ with the discrete topology, and the subset $T \subseteq B(\mathfrak m)$ consisting of all points $\{s_i\}$ such that $\{U_{s_i}\}_{i=1}^\infty$ is a base at a point $x\in X$; assign the point $x \in X$ to the point $\{s_i\}\in T$.
(b) (Michael [1971a]) Show that every first-countable space $X$ is a continuous image of a first-countable Hausdorff space under an open mapping, and deduce that in (a) the assumption that $X$ is a $T_0$-space can be omitted.
Hint (Shimrat [1956]). Define in $B(\mathfrak m)$, where $m = |X|$, a family $\{A_x\}_{x\in X}$ of pairwise disjoint dense subsets and consider the subset $\bigcup_{x\in X}(\{x\}\times A_x)$ of the Cartesian product $X \times B(\mathfrak m)$.
Remark. The construction in the above hint shows that every topological space is a continuous image of a Hausdorff space under an open mapping. Isbell proved more in [1969]: every topological space is a continuous image of a hereditarily paracompact and hereditarily strongly zero-dimensional space under an open mapping. Further information related to parts (a) and (b) can be found in Junnila [1978] and R. Pol [1981].
(c) (Arhangelskii [1963], Franklin [1965]) Observe that the construction outlined in Exercise 2.4.G(b) shows that sequential spaces can be characterized as the images of metrizable spaces under quotient mappings and Frechet spaces can be characterized as the images of metrizable spaces under hereditarily quotient mappings.

• Arkhangel’skij, A. V., Bicompact sets and the topology of spaces, Trans. Mosc. Math. Soc. 13, 1-62 (1965); translation from Tr. Mosk. Mat. O.-va 13, 3-55 (1965). ZBL0162.26602, MR0150733.
• Franklin, S. P., Spaces in which sequences suffice, Fundam. Math. 57, 107-115 (1965). ZBL0132.17802, MR180954.
• Isbell, J. R., A note on complete closure algebras, Math. Syst. Theory 3, 310-312 (1969). ZBL0182.34303, MR252296.
• Junnila, H. J. K., Stratifiable pre-images of topological spaces, Topology, Vol. II, 4th Colloq. Budapest 1978, Colloq. Math. Soc. Janos Bolyai 23, 689-703 (1980). ZBL0437.54019, MR588817.
• Michael, Ernest A., On representing spaces as images of metrizable and related spaces, General Topology Appl. 1, 329-343 (1971). ZBL0227.54009, MR293604.
• Ponomarev, V., Axioms of countability and continuous mapping, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 8, 127-134 (1960). ZBL0095.16301, MR0116314; in Russian.
• Pol, Roman, On category-raising and dimension-raising open mappings with discrete fibers, Colloq. Math. 44, 65-76 (1981). ZBL0479.54008, MR633099.

I will add that the same exercises is mentioned in the answer to this question on Mathematics Stack Exchange: Every Tychonoff space is an image of a Moscow space under a continuous open mapping. The references in that answer might be worth looking at, too.

¹This older edition used to be available online - I do not know whether it is still accessible somewhere. Here is a Wayback Machine snapshot of the website where it used to be.