We introduced closed sets and clopen sets. We'll visit two definitions of closed sets. First, a set is closed if it is the complement of some open set, and second, a set is closed if it contains all of its limit points. We see examples of sets both closed and open (called "clopen sets") and we see examples of sets that are neither closed nor open. We complete several quick proofs that a set is closed. We see why the empty set is closed, and why the real numbers are closed. #realanalysis
Open Sets: https://youtu.be/pnWgj8jjs3w
Proof Union of Open Sets is Open: https://youtu.be/5tiKwVmEglA
Limit Points: https://youtu.be/RmsvftFNMtE
Proof for Equivalent Definitions of Closed Sets: https://youtu.be/Mh1noyFVNVg
Real Analysis Course: https://www.youtube.com/playlist?list=PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli
Real Analysis exercises: https://www.youtube.com/playlist?list=PLztBpqftvzxXAN05Gm3iNmpz9SkVfLNqC
Table of Contents
00:00 Intro
00:29 Defining Closed Sets with Open Sets
01:34 Closed Set Examples
04:40 Defining Closed Sets with Limit Points
05:59 More Closed Set Examples
06:48 A Set Neither Closed Nor Open
08:58 Closure of a Set
09:00 Showing Closed Intervals are Closed Sets Using Limit Point Definition
◉Textbooks I Like◉
Graph Theory: https://amzn.to/3JHQtZj
Real Analysis: https://amzn.to/3CMdgjI
Proofs and Set Theory: https://amzn.to/367VBXP (available for free online)
Statistics: https://amzn.to/3tsaEER
Abstract Algebra: https://amzn.to/3IjoZaO
Discrete Math: https://amzn.to/3qfhoUn
Number Theory: https://amzn.to/3JqpOQd
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