We derive the product topology based on the universal property we want it to fulfill. We then see how to form the product of continuous maps and list the most important properties of product spaces. In particular, we show that the product of topological manifolds is again a topological manifold.
00:00 Introduction
00:17 Derivation of the Product Topology
20:18 Definition: Product Topology
24:17 Characteristic Property of Product Topology
34:45 Definition: Product of Continuous Maps
38:04 Prop: Product of continuous maps is continuous
42:12 Prop: Product of homeomorphisms is homeomorphism
48:31 Properties of Product Spaces
52:09 Cor: Product of topological manifolds is a topological manifold
This lecture follows Lee's "Introduction to topological manifolds", chapter 3.
A playlist with all the videos in this series can be found here:
https://www.youtube.com/playlist?list=PLd8NbPjkXPliJunBhtDNMuFsnZPeHpm-0