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

[@curiouslockpicker8971]

Love the channel! One quick correction: 7 is not a limit point of {7}, at least according to the definition given in Understanding Analysis.

[@WrathofMath]

Thanks Curious Lock Picker, I've been meaning to pin a comment about that on this video. Somehow I had it in my head while filming this that isolated points are trivial limit points, which they definitely are not. Correction for the reading viewer: The two common definitions of a limit point of a set A are 1) a limit of a sequence in A and 2) a point (not necessarily in A) such that every epsilon neighborhood around it contains some element of A other than itself. There is almost certainly no definition which would classify 7 as a limit point of {7}.

[@curiouslockpicker8971]

@@WrathofMath Cheers! After analysis, I'm going to study abstract algebra and will definitely use your videos to supplement and strengthen what I read.

[@okikiolaotitoloju2208]

That's what I thought also, as a limit point has to be such that it's not an element of the sequence of points from the set.

[@Bedoroski]

@@WrathofMath Wait, since 7 isn't a limit point of our singleton set, it implies that this set has no limit point. Would it still mean that our set is closed, given it has no limit point to start with and thus a vacuous truth?

[@samuelhawksworth7719]

I'm studying point set topology and hadn't been taught compactness in real analysis before (since it's covered in topology as an optional module) and I really struggled getting to grasps with it. its nice to see you go through plenty of examples in the euclidean topology that can act as a referral point. So thank you!

[@soumyadipghosh6925]

nailed 5, maybe more of these? These kinds of examples are really indispensable. Thanks a ton

[@WrathofMath]

Glad to help! I agree, examples are critical for a lot of analysis topics.

[@TranquilSeaOfMath]

Nice examples and discussions of reasons.

[@summatim]

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 (again)

[@WrathofMath]

Thanks for the timestamps!

[@Bedoroski]

Great video done with dedication

[@WrathofMath]

Thank you!

[@okikiolaotitoloju2208]

Hey, love the examples!! Just wanted to ask where is the video that explains that a set is compact if it is closed and bounded?

[@WrathofMath]

Thank you! I am still working on it. It pains me to say in the video you can check it out when it isn’t done yet, but I just want the videos to be future proof and not waste time in the video saying “it’s not out now, but maybe if you’re watching five months after the day this video was published it might be out, check the description”. It will all be here eventually!

[@okikiolaotitoloju2208]

@@WrathofMath Ok no worries, if that's the case could you recommend me to another channel which has a video explaining that a set is compact iff it is closed and bound (using open covers), I've been looking but can't find any.

[@WrathofMath]

@@okikiolaotitoloju2208 I don't know a channel with the proof offhand, but I would certainly recommend the textbooks I'm primarily using for this playlist: Understanding Analysis by S. Abbott and Real Analysis by Jay Cummings. I think Jay's book has the proof.

[@CliveDhlalambi]

perfect 😉

[@WrathofMath]

Thank you!