Identifying Open, Closed, and Compact Sets | Real Analysis Exercises

We look at six sets and determine if each one is either open, closed, compact, or any combination of the three. We will use multiple definitions in our solutions. An open set is a set whose elements all have some neighborhood around them completely contained in the set. A closed set is a complement of an open set, or equivalently a set whose limit points are all contained in the set. A compact set is closed and bounded, or equivalently a set whose open covers all have finite subcovers. #realanalysis Thanks Summatin for the chapters! 0:00 Intro 0:20 Integers 2:07 Union of reciprocals of natural numbers with set containing zero 3:20 Real numbers 4:05 Union of open and closed intervals 5:01 Rational numbers 5:42 Singleton set containing only seven 6:23 Real numbers revisited Open Sets: https://youtu.be/pnWgj8jjs3w Closed Sets: https://youtu.be/O3ECYp4ICFA Open Covers, Finite Subcovers, and Compact Sets: https://youtu.be/dAz9ax-bjzU Real Analysis Course: https://www.youtube.com/playlist?list=PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli Real Analysis exercises: https://www.youtube.com/playlist?list=PLztBpqftvzxXAN05Gm3iNmpz9SkVfLNqC ◉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 ★DONATE★ ◆ Support Wrath of Math on Patreon for early access to new videos and other exclusive benefits: https://www.patreon.com/join/wrathofmathlessons ◆ Donate on PayPal: https://www.paypal.me/wrathofmath Thanks to Petar, dric, Rolf Waefler, Robert Rennie, Barbara Sharrock, Joshua Gray, Karl Kristiansen, Katy, Mohamad Nossier, and Shadow Master for their generous support on Patreon! Thanks to Crayon Angel, my favorite musician in the world, who upon my request gave me permission to use his music in my math lessons: https://crayonangel.bandcamp.com/ Follow Wrath of Math on... ● Instagram: https://www.instagram.com/wrathofmathedu ● Facebook: https://www.facebook.com/WrathofMath ● Twitter: https://twitter.com/wrathofmathedu My Math Rap channel: https://www.youtube.com/channel/UCQ2UBhg5nwWCL2aPC7_IpDQ/featured