Lecture 3: Compact Sets in Rⁿ

MIT 18.S190 Introduction To Metric Spaces, IAP 2023
Instructor: Paige Bright
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We motivate the concept of compact sets on Euclidean space with norms, support of functions, and finite sets. We also prove the Heine-Borel theorem and introduce the notions of topological and sequential compactness.
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[@weamah4923]

Thank you for the lecture! I have one remark about something mentioned around minute 45 though. The "least upper bound property of R" assumes not only boundedness, but also non-emptiness. So to prove that { \0 \leq c \leq 1 , [0,c] has a finite subcover \} admits a supremum, one must first prove that it's non-empty. This can be done by proving that 0 belongs to the set; this is true because any open cover of {0} must contain an open set containing zero which, on its own, constitues a (finite) subcover of {0}.

[@shawan7089]

Please make a course on number theory and abstract algebra

[@sebon11]

First one !!!!

[@cycklist]

Please find some meaning in your life.