In this chapter we prove some results on topological spaces that will be needed later and go beyond the basic topological results and notions assembled in the Appendix Chap. 12. The chapter consists of two independent parts.
In the first part (Sects. 1.1–1.3) we introduce, after a quick review of countability properties, paracompact spaces. This is one of the central topological notions in this book. We show that the following classes of topological spaces are paracompact: Metrizable spaces (Proposition 1.13) and locally compact, second countable Hausdorff spaces (Proposition 1.10), see also Remark 1.14. Then we show that paracompact Hausdorff spaces are normal (Proposition 1.18). Hence Urysohn’s separation theorem, the Tietze extension theorem (Theorem 1.15), and the shrinking lemma (Proposition 1.20, Corollary 1.21) are available for paracompact spaces.
The second part (Sects. 1.4 and 1.5) introduces relative versions of Hausdorff spaces and compact spaces: separated maps and proper maps.