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Topological Preliminaries

Chapter
Part of the Springer Studium Mathematik - Master book series (SSMM)

Abstract

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.

Keywords

Topological Space Compact Space Relative Version Hausdorff Space Metrizable Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Springer Fachmedien Wiesbaden 2016

Authors and Affiliations

  1. 1.Technische Universität DarmstadtDarmstadtGermany

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