Basic Topology of R $$\mathbf{R}$$

Abstract

What follows is a fascinating mathematical construction, due to Georg Cantor , which is extremely useful for extending the horizons of our intuition about the nature of subsets of the real line. Cantor’s name has already appeared in the first chapter in our discussion of uncountable sets. Indeed, Cantor’s proof that R\mathbf{R} is uncountable occupies another spot on the short list of the most significant contributions toward understanding the mathematical infinite. In the words of the mathematician David Hilbert, “No one shall expel us from the paradise that Cantor has created for us.”