MATH 242: Analysis 1 (McGill Final)

Currently Playing: All About Closed Sets and Closures of Sets (and Clopen Sets) | Real Analysis

Support the production of this course by joining Wrath of Math to access all my real analysis videos plus the lecture notes at the premium tier! https://www.youtube.com/channel/UCyEKvaxi8mt9FMc62MHcliw/join 🛍 Get the coolest math clothes in the world! https://mathshion.com/ Real Analysis course: https://www.youtube.com/playlist?list=PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli Real Analysis exercises: https://www.youtube.com/playlist?list=PLztBpqftvzxXAN05Gm3iNmpz9SkVfLNqC Get the textbook! https://amzn.to/45kcMjq 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 ★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 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 0:00 Intro 0:29 Defining Closed Sets with Open Sets 1:34 Closed Set Examples 4:40 Defining Closed Sets with Limit Points 5:59 More Closed Set Examples 6:48 A Set Neither Closed Nor Open 8:58 Closure of a Set 9:00 Showing Closed Intervals are Closed Sets Using Limit Point Definition