Scattered spaces and their compactifications

We show that a known restriction on the cardinalities of closures of subspaces of scattered We then find a wide class of spaces, $\text{∥}\text{A}\text{̄}\text{∥ ⩽ 2}{}^{\mathrm{\parallel A\parallel }}$, cannot be improved to $\text{∥}\text{A}\text{̄}\text{∥ ⩽ ∥A∥}{}^{\mathrm{\lambda }}$, for any λ.T.3$\text{1}\text{2}$. scattered spaces which have no scattered compactification: these spaces are derived from regular filters over cardinals bigger than N₁.