This is a joint work with Evgenii Reznichenko.

A topological group is said to be $\mathbb{R}$-factorizable if, given any continuous function $f:G\to \mathbb{R}$, there exists a continuous homomorphism $h:G\to H$ to a second-countable topological group $H$ and a continuous function $g:H\to \mathbb{R}$ such that $f=g\circ h$. The main unsolved problems of the theory of $\mathbb{R}$-factorizable groups are as follows:

1. Is the property of being an $\mathbb{R}$-factorizable group topological? In other words, is any topological group homeomorphic to an $\mathbb{R}$-factorizable one $\mathbb{R}$-factorizable?

2. Is the square of an $\mathbb{R}$-factorizable group $\mathbb{R}$-factorizable?

3. Is any $\mathbb{R}$-factorizable group pseudo-${\mathrm{\aleph }}_1$-compact, that is, contains no uncountable locally finite family of open sets?

4. Is the image of an $\mathbb{R}$-factorizable group under a continuous homomorphism $\mathbb{R}$-factorizable?

We show that if the answer to question 2 is positive, then so is the answer to question 1. Also, if the answer to question 4 is positive, then so is the answer to question 3, and if the answer to question 3 is negative, then so are the answers to questions 1 and 2. Note that there are examples of $\mathbb{R}$-factorizable groups $G$ and $H$ such that $G\times H$ is not $\mathbb{R}$-factorizable.

Our main concern is the pseudo-${\mathrm{\aleph }}_1$-compactness of $\mathbb{R}$-factorizable groups. We prove that an $\mathbb{R}$-factorizable group $G$ is pseudo-${\mathrm{\aleph }}_1$-compact if it satisfies any of the following conditions:

(i) the weight of $G$ is at most ${\omega }_1$;

(ii) the pseudocharacter of $G$ equals ${\omega }_1$;

(iii) $G^2$ is $\mathbb{R}$-factorizable;

(iv) $G$ contains a nonmetrizable compact subspace;

(v) $G$ contains a Lindel"of subspace of uncountable pseudocharacter.