Is every Hausdorff sequential space monolithic? A space $X$ called monolithic if for any subset $A$ with $|A|\le \kappa$, then $nw(\overline{A})\le \kappa$. Is every Hausdorff sequential space monolithic? What I have tried: For any subset $A$, because $nw(\overline{A})\le|\overline{A}|\le [A]^\omega=\kappa$, then it is true. Am I right?

The Mrówka space $\Psi$ is a counterexample: it’s even first countable, but it’s a separable space with net weight $2^\omega$. Your mistake is in thinking that $\left|[A]^\omega\right|$ must equal $|A|$. Suppose that $A$ is countably infinite: then $\left|[A]^\omega\right|=2^\omega$, not $\omega$.