Is there a normal separable sequential $\aleph$-space with uncountable extent?

It is a classical fact from the undergraduate course of General Topology that under CH (more precisely, under $$2^{\omega_1}>\mathfrak c$$) every separable normal space has countable extent, i.e., does not contain uncountable closed discrete subspaces.

On the other hand, MA$$+\neg$$CH there are examples which are separable, normal, and have uncountable extent. One of such examples is the Mrowka-Isbell space $$\Psi(\mathcal A)$$ for a suitable uncountable almost disjoint family $$\mathcal A$$. The Mrowka-Isbell space is even a $$\sigma$$-space, i.e., has a $$\sigma$$-discrete network.

I am interested in finding such counterexamples among sequential $$\aleph$$-spaces.

Let us recall that a regular topological space $$X$$ is an $$\aleph$$-space if it has a $$\sigma$$-discrete $$k$$-network.

A family $$\mathcal N$$ of subsets of a topological space $$X$$ is called a $$k$$-network if for any open set $$U\subset X$$ and any compact subset $$K\subset U$$ there exists a finite subfamily $$\mathcal F\subset\mathcal N$$ such that $$K\subset\bigcup\mathcal F\subset U$$.

Problem. Is there an example of a normal separable sequential $$\aleph$$-space with uncountable extent?

Remark. Such space, if exists, should have uncountable sequential order. In this case also $$2^{\omega_1}=\mathfrak c$$.

• The classical fact is known as Jones' lemma as you probably know. – Henno Brandsma Feb 2 at 14:51