Topology Lecture 09: Countability Properties

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:    • Topology