The height and width of a topological space

The notion of Cantor-Bendixon derivatives of a topological space is as old as the discovery of mathematical infinity. The derivatives of a topological space introduce a topological invariant in terms of cardinals. A topological space is called scattered if the derivatives eventually vanish, i.e., a derived subspace in the Cantor-Bendixon process turns to be the empty set. We shall mention some known results about the cardinal sequence of a topological space and then discuss this notion in the context of Boolean algebras. Finally, we shall end the talk by discussing the importance of a major open problem about thin-tall locally compact scattered Hausdorff topological spaces.