Countable network vs countable Borel network

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

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

$$\bullet$$ a Borel network if $$\mathcal N$$ 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 $$\mathbb R$$).

• Do you have an example of a space with a countable network but not a countable closed network? – James Hanson Jan 9 at 16:12
• @JamesHanson The two-point space $X=\{a,b\}$ with topology $\{\emptyset,\{a\},X\}$ has a countable open network but does not have countable closed network. But it is not $T_1$-space. Now I will think on a $T_1$-example. – Taras Banakh Jan 9 at 16:33
• @JamesHanson The real line endowed with the cofinite topology has a countable network but does not have a countable closed network (since the closure of any infinite set is the whole line). This is a $T_1$-example, but it is not Hausdorff. – Taras Banakh Jan 9 at 16:38
• @JamesHanson A Hausdorff example can be constructed as follows: Let $\tau_E$ be the standard Euclidean topology on the real line and $\tau$ be the topology generated by the subbase $\tau_E\cup\{\mathbb R\setminus \mathbb Q\}$. The real line endowed with the topology $\tau$ is a functionally Hausdorff second-countable space, which does not have a countable closed network (to prove this fact, apply the Baire Theorem). – Taras Banakh Jan 9 at 16:52