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:

[MortyInARobe]

these videos are so amazing i really hope more people get to see them, they have really nice explanations and this has been an amazing way to learn topology!

[dutonic]

the textbook I'm using says the open cover only needs to contain the space X. Not that it needs to equal it.

[saraperestrelo]

I am following your videos to understand Topology for the first time. Very good explanations.
My question:
In a discrete space, why is every subset open? It's a question also from the first lecture, I didn't understand. Why can't we find a closed subset? Ex: a set of points.

[javierromanjimenez]

I am looking forward to your new videos

[Acheron]

If you don't mind me asking, what app do you use for writing and recording?