This is a joint work with Evgenii Reznichenko.

A topological group is said to be $\mathbb R$-factorizable if, given any continuous function $f\colon G\to \mathbb R$, there exists a continuous homomorphism $h\colon G \to H$ to a second-countable topological group $H$ and a continuous function $g\colon H\to \mathbb R$ 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-$\aleph_1$-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-$\aleph_1$-compactness of $\mathbb R$-factorizable groups. We prove that an $\mathbb R$-factorizable group $G$ is pseudo-$\aleph_1$-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.

Slides: https://drive.google.com/file/d/13J5f5EwTvjgO3NQ4oSYoB3uPlxGoE3i6/view?usp=sharing