15 - Special Topics

Summary

This chapter highlights connections of the book’s topics to structures used in all areas of mathematics. Cantor famously proved that no set can be mapped onto its power set. We present some analogous results for metric spaces and posets. On the category of topological spaces, we consider endofunctors built from the Vietoris endofunctor using products, coproducts, composition, and constant functors restricted for Hausdorff spaces. Every such functor has an initial algebra and a terminal coalgebra. Similar results hold for the Hausdorff functor on (complete) metric spaces. Extending a result of Freyd, we exhibit structures on the unit interval [0, 1] making it a terminal coalgebra of an endofunctor on bipointed metric spaces. The positive irrationals and other subsets of the real line are described as terminal coalgebras or corecursive algebras for some set functors, calling on results from the theory of continued fractions.