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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 $2^{\omega_1}=\mathfrak c$.

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

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