We define first and second countability, show that euclidean space is second countable, and explain that the topology on a first countable space can be characterized in terms of convergent sequences.
00:00 Introduction
00:31 Definition: Second Countability
01:00 Example: Euclidean Space
17:01 Definition: Neighborhood basis at a point
17:44 Definition: First Countability
18:08 Prop: Every second countable space is first countable
21:54 Prop: Characterizations of closure and interior by sequences in first countable spaces
This lecture follows Lee's "Introduction to topological manifolds", chapter 2.
A playlist with all the videos in this series can be found here:
https://www.youtube.com/playlist?list=PLd8NbPjkXPliJunBhtDNMuFsnZPeHpm-0
