Just as the title explains, is the Sorgenfrey Line monotonically monolithic [(see the definition)](https://math.stackexchange.com/questions/297948/what-is-the-relation-between-kappa-monolithic-and-monotonically-monolithic)?



Is the Sorgenfrey Line monotonically monolithic?
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.


