By the Gelfand duality, the theory of C*-algebras can be regarded as "non-commutative topology". In a joint work with Piotr Koszmider at IMPAN, we investigated the non-commutative analogues of the scattered spaces, parallel to the classical research in set-theoretic topology. The so called scattered C*-algebras, despite being around in the literature, have not been subject to the tools from set-theoretic topology. The techniques and constructions of compact, Hausdorff scattered spaces, or equivalently (by the Stone duality) superatomic Boolean algebras, have already led to many fundamental results in the theory of Banach spaces of the form C(K), or more generally Asplund spaces. In fact scattered C*-algebras were introduced as C*-algebras which are Asplund as Banach spaces. I will introduce the notion of the Cantor-Bendixson derivatives for these C*-algebras, and present some of the basic properties of such algebras. I will also show how it can be used to construct C*-algebras with exotic properties, which are non-commutative versions of well-known scattered spaces. In particular, the constructions of non-commutative Psi-spaces and thin tall spaces lead to new discoveries about the preservation of the "stability" for non-separable C*-algebras.