Replicable Rooms and Boundary Shrinkable Skyscrapers

Abstract

‘Replicable Rooms and Boundary Shrinkable Skyscrapers’ points out precisely which properties of skyscrapers are required in the remainder of the proof of the disc embedding theorem. To achieve this, it introduces an abstraction of towers, known as buildings. The required properties for a generalized skyscraper include boundary shrinkability and replicability. The former allows the conclusion that the vertical boundary of a generalized skyscraper is a solid torus. Replicability ensures that any generalized skyscraper contains uncountably many other skyscrapers as subsets. Both of the above properties will be essential in the construction of the design in a subsequent chapter.