CBB and CBA spaces are metric spaces with curvature bounded below and above, respectively, in the sense of Alexandrov. Extremal subsets are singular sets in CBB spaces introduced by Perelman and Petrunin, which are defined by a purely metric condition but also closely related to the topological structure of CBB spaces.

This talk is divided into two parts. The first half concerns extremal subsets and collapsing in CBB geometry. A CBB space is said to collapse if it is sufficiently close to a CBB space of lower dimension in the Gromov-Hausdorff distance. It is expected that a collapsing space admits a singular fibration structure over its limit space, whose singular fibers arise over extremal subsets of the limit space. We will describe such a relationship between a collapsing space and its limit space in terms of algebraic topology.

The second half deals with extremal subsets in GCBA geometry. The structures of CBB and CBA spaces are quite different in general. However, if we restrict our attention to geodesically complete CBA spaces, i.e., GCBA spaces, there are many parallels between CBB and GCBA geometries. From this point of view, we will introduce the notion of extremal subsets in GCBA spaces and discuss their properties, especially their connection with the topological structure of GCBA spaces.