The passage from commutative to noncommutative mathematics has stimulated a big part of mathematical research since the mid 20th century which resulted the unprecedented impact of this programme on the shape of today's mathematics. Quite central in it were the noncommutative geometry and topology but a noncommutative set-theoretic topology was not properly developed. This situation is changing. Perhaps the crucial momentum was given by the independence results concerning an important noncommutative object, the Calkin algebra (noncommutative analogue of betaN-N) due to Phillips, Weaver (2007) and Farah (2011).
In this talk I we will try to explain some of the challenges of the noncommutative set-theoretic topology focusing on the noncommutative analogues of scattered compact or locally compact Hausdorff spaces known as scattered C*-algebras. These objects have been investigated since the 80ties but in separation from the classical commutative case (ordinals, Psi-spaces, ladder system spaces, thin-(very) tall spaces etc). For example, no analogue of the Cantor-Bendixson derivative was used.
The talk is based on a joint project with Saeed Ghasemi (IM PAN) in which we developed the noncommutative Cantor Bendixson derivative and looked at the basic constructions and problems corresponding to the classical programme of 'cardinal sequences' for scattered compact spaces (or equivalently superatomic Boolean algebras).
The only background of the audience in noncommutative issues which will be assumed concerns the multiplication of matrices.
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