The Heine–Borel and Baire Category Theorems

Abstract

This chapter starts with the Heine–Borel theorem and its characterization of complete orders, and then uses Borel’s theorem to give a measure-theoretic proof that $\mathbf{R}$ is uncountable. Other topics focus on measure and category: Lebesgue measurable sets, Baire category, the perfect set property for $\mathcal{G}_{\delta }$ sets, the Banach–Mazur game and Baire property, and the Vitali and Bernstein constructions.