Topology Lecture 02: Closed Sets & Closure

We define closed sets and give some examples. Then we learn about the closure, interior, exterior, and boundary of sets. We conclude by proving several characterizations for these objects. 00:00 Introduction 00:21 Review 02:04 Definition: Closed Set 04:27 Consequences of the Definition 09:49 Example: Closed Interval 10:36 Example: Closed Ball 15:16 Example: Subset of Discrete Space 16:16 Definition: Closure 19:01 Definition: Interior 20:52 Definition: Exterior 22:04 Definition: Boundary 24:36 Prop: Characterization of Interior 27:15 Prop: Characterization of Boundary 28:22 Prop: Characterization of Closure 33:00 Prop: Closure is the Union of Interior and Boundary 34:18 Prop: Interior is open, Closure is closed 34:58 Prop: Characterization of Open Sets 40:19 Prop: Characterization of Closed Sets This lectrure 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