Lecture 3: Compact Sets in Rⁿ

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MIT 18.S190 Introduction To Metric Spaces, IAP 2023
Instructor: Paige Bright

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.

License: Creative Commons BY-NC-SA

  1. MIT OpenCourseWare
  2. Bolzano-Weierstrass
  3. DubbedWithAloud
  4. Heine-Borel theorem
  5. Normed spaces
  6. analysis on finite sets
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Комментарии
Автор

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}.

weamah
Автор

Please make a course on number theory and abstract algebra

shawan
Автор

it is not true in general that every metric space that is closed and bounded is compact

EastBurningRed