A topological group is embeddble in a product of a family of second-countable topological groups if and only if it is $\omega$-narrow

Question

How to prove the following property: a topological group is topologically isomorphic to a subgroup of the product of some family of second-countable topological groups if and only if it is ω-narrow

Answer 1

This is a result of my first supervisor Igor Y. Guran. Its proof it rather long and can be found, for instance, in a book “Topological groups and related structures” by his supervisor Alexander V. Arhangel'skii and co-student Mikhail G. Tkachenko (Atlantis Press, Paris; World Sci. Publ., NJ, 2008), where this results is formulated as Theorem 3.4.23.