A topological space $X$ is called *analytic* if it is a continuous image of a Polish space, i.e., the image of a Polish space $P$ under a continuous surjective map $f:P\to X$.

We say that a topological space $X$ is a *Borel image of a Polish space* if $f(P)=X$ for some Borel surjective function $f:P\to X$ defined on a Polish space $P$.

We recall that a function $f:X\to Y$ between topological spaces is *Borel* if the preimage $f^{-1}(U)$ of any open set $U\subset Y$ is Borel in $X$.

Problem 1.Is a (regular) topological space $X$ analytic if it is a Borel image of a Polish space $P$? Will be the answer affirmative if $X$ is a (Baire) topological group?

The following theorem reduces the problem to finding a countable network for the space $X$.

Let us recall that a family $\mathcal N$ of subsets of a topological space $X$ is a *network* for $X$ if for any open set $U\subset X$ and point $x\in U$ tehre exists a set $N\in\mathcal N$ such that $x\in N\subset U$.

Theorem.A regular topological space $X$ is analytic if and only if $X$ has a countable network and $X$ is a Borel image of a Polish space.

*Proof.* Let $\mathcal N$ be a countable network for $X$. Since $X$ is regular, we can replace each set $N\in\mathcal N$ by its closure $\bar N$ and assume that $\mathcal N$ consists of closed subsets of $X$.

Let $f:P\to X$ be a Borel surjective map defined ona Polish space. It follows that $\{f^{-1}(N):N\in\mathcal N\}$ is a countable family of Borel subsets of $P$. By Exercise 13.5 in the "Classical Descriptive Set Theory" of Kechris, there exists a bijective continuous map $g:Z\to P$ from a Polish space $Z$ such that each set $f^{-1}(N)$, $N\in\mathcal N$, is clopen. Then the map $f\circ g:Z\to X$ is continuous, witnessing that $X$ is an analytic space. $\square$

Therefore, Theorem reduces Problem 1 to the following equivalent

Problem 2.Assume that a regular space $X$ is Borel image of a Polish space. Has $X$ a countable network?

I can only prove (using the Four Poles Theorem) that $X$ has *countable spread* (which means that $X$ each discrete subspace of $X$ is countable).

Problem 3.Assume that a (regular) topological space $X$ is a Borel image of a Polish space.

- Is $X$ hereditarily Lindelöf?

- Is $X$ hereditarily separable?

- Is $X\times X$ a Borel image of a Polish space?

**Added in Edit.** Under PFA the answer to Problem 3(1) is affirmative (because PFA implies that each regular space with countable spread is hereditarily Lindelof, see Theorem 8.10 in Todorcevic's book "Partition Problems in Topology").