Topology Lecture 21: Compactness I

We define compactness in terms of open covers and see several basic examples. We then prove that compactness is preserved by continuous functions. 00:00 Introduction 00:47 Definition: Open cover 05:22 Definition: Compactness 09:24 Examples of compact spaces 15:11 Compact subspace lemma 29:17 Convergent sequence is compact as subspace 36:52 Theorem: Continuous images of compact spaces are compact This lecture follows Lee's "Introduction to topological manifolds", chapter 4. A playlist with all the videos in this series can be found here:    • Topology