All About Closed Sets and Closures of Sets (and Clopen Sets) | Real Analysis
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: • Intro to Open Sets (wi...
Proof Union of Open Sets is Open: • Proof for Unions and I...
Limit Points: • Limit Points (Sequence...
Proof for Equivalent Definitions of Closed Sets: • A Set is Closed iff it...
Real Analysis Course: • Real Analysis
Real Analysis exercises: • Real Analysis Exercises
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
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Пікірлер: 12
NOTE: Around 08:58 when we discuss the closure of a set, I must clarify that the closure of A being a closed set is not as trivial as I made it sound. Certainly the closure of A contains the limit points of A, but when unioning A with L, it is possible we introduced NEW limit points of the set L, and thus to show the closure of A is closed, we'd need to consider this possibility. As it turns out, new limit points will not be introduced in this process, so closures are indeed closed, but again - it's not as trivial as I made it sound.
5:31 I like that you connected your discussion here with closure under an operation.
Bit late on this (I am just starting Uni), but these are amazing and so much better than lectures. You are doing God's work my friend. Keep pumping more high level math videos. I will watch all of them.
Your lectures have definitely helped me a lot in these proofs in discrete maths
@WrathofMath
Жыл бұрын
Glad to hear it! Thanks for watching and let me know if you have any video requests!
Thank you. it was great❤🔥
@WrathofMath
11 ай бұрын
Glad to help - thanks for watching!
Legend man, legend man
@WrathofMath
7 ай бұрын
Appreciate ya
Saving lives...
@WrathofMath
Жыл бұрын
I'm doing my best! Let me know if you have any questions or video requests!
@mfonpeter124
4 ай бұрын
Will it be possible to create lectures on Topology in Real Numbers with n dimensions😣. @@WrathofMath