This question was motivated by my answer to this MO question, which asked about the characterization of spaces that belong to the smallest class of topological spaces that is closed under taking subspaces, quotient spaces and contains all Euclidean spaces $\mathbb R^n$. It turns out that this class coincides with the class of subspaces of sequential spaces of cardinality $\le\mathfrak c$ possessing a countable $k$-network.
So, the original problem has reduced to the problem of recognizing subspaces of sequential spaces with countable $k$-network.
Let us recall that a topological space is called subsequential if it is homeomorphic to a subspace of a sequential topological space.
A family $\mathcal N$ of subsets of a topological space $X$ is called a $k$-network if for each open set $U\subset X$ and compact subset $K\subset U$ there exists a finite subfamily $\mathcal F\subset\mathcal N$ such that $K\subset\cup\mathcal F\subset U$.
In the theory of generalized metric spaces, regular spaces with countable $k$-network are called $\aleph_0$-spaces.
Question 1. Let $X$ be a subsequential space with a countable $k$-network. Is $X$ a subspace of a sequential space with a countable $k$-network?
I suggest that the answer to this question is negative and a suitable counterexample is constructed in Theorem 7.3 of this paper of Franklin and Rajagopalan, but I cannot prove that the countable subsequential space $PS$ of subsequential order $\omega_1$ constructed in this theorem can (or cannot) be embedded into a sequential spaces with countable $k$-network.
Another possible counterexample to Question 1 can be the countable power $S^\omega_\omega$ of the Frechet-Urysohn fan $S_\omega$. We recall that $S_\omega$ is the quotient space of $S_1\times\mathbb N/\{0\}\times\mathbb N$, where $S_1=\{0\}\cup\{\frac1n\}_{n=1}^\infty$ is the convergent sequence.
Question 2. Is the space $S^\omega_\omega$ subsequential? If yes, does it embed into a sequential space with a countable $k$-network?
