Topology Lecture 22: Compactness II

We prove important properties about compact subsets.

00:00 Introduction
01:40 Compact subsets of hausdorff spaces can be separated by open sets.
18:51 Tube lemma
33:09 Closes subsets of compact spaces are compact
38:48 Compact subsets of hausdorff spaces are closed
43:27 Compact subsets of metric spaces are bounded
49:13 Finite products of compact spaces are compact
57:26 Quotients 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:

[ChrisRossaroGG]

12:35 I was trying to prove the lemma by my self, wondering why I should take a finite union of open sets U_p_i (actually a union was also an open set). Then I thought about the intersection of sets V_p_i, which is open only if finite and everything became clear.