Is the Sorgenfrey Line monotonically monolithic?

Question

Just as the title explains, is the Sorgenfrey Line monotonically monolithic (see the definition)?

Answer 1

No, the Sorgenfrey line $\mathbb{R}_l$ is not even $\aleph_0$-monolithic.

Note that any network $\mathcal{N}$ on $\mathbb{R}_l$ must have at least $2^{\aleph_0}$ elements, since for each $x \in \mathbb{R}_l$ there is an $N_x \in \mathcal{N}$ such that $x \in N_x \subseteq [x,x+1)$. On the other hand $\mathbb{R}_l$ is separable since $\mathbb{Q}$ remains dense in the Sorgenfrey topology. So $\aleph_0 =d(\mathbb{R}_l) \lt nw(\mathbb{R}_l)$ and hence $\mathbb{R}_l$ is not $\aleph_0$-monolithic.

By the way: monotonically monolithic $\implies$ monolithic $\implies$ $\aleph_0$-monolithic.