Definition.A family $\mathcal N$ of subsets of a topological space $X$ is called$\bullet$ a

networkif for any open set $U\subset X$ and point $x\in U$ there exists a set $N\in\mathcal N$ such that $x\in N\subset U$;$\bullet$ a

closed networkif $\mathcal N$ is a network whose every set $N\in\mathcal N$ is closed in $X$;$\bullet$ a

Borel networkif $\mathcal N$ is a network whose every set $N\in\mathcal N$ is a Borel subset of $X$.

Since each base of the topology is a network, each second countable space has a countable Borel network. On the other hand, a regular topological space has a countable network if and only if it has a closed (and hence Borel) network.

Problem.Is there a (desirably Hausdorff) topological space $X$ that has a countable network but does not have a countable Borel network?

**Remark.** Such space $X$, if exists, cannot be second countable or regular.

**Added in Edit.** Alex Ravsky informed me that the real line endowed with the cofinite topology $\tau=\{\emptyset\}\cup\{\mathbb R\setminus F:F$ is finite$\}$ is an example of a $T_1$-space with a countable network that does not have a countable Borel network (since each Borel subset of $(\mathbb R,\tau)$ is either countable or has countable complement in $\mathbb R$).