Let $X$ be a topological space. For any set $S$, let us denote its cardinality by $|S|$. The weight of $X$ is defined by $$ w(X) = \inf\{|\mathcal B|: \mathcal B\mbox{ is a basis for the topology of $X$}\}. $$ And let us recall that a topological space is called scattered if, and only if, every subspace $A\subset X$ has an isolated point (with respect to $A$).
My question is whether there exists a compact scattered space $X$ such that $ w(X) < |X|. $