Math 441 - 3.1 and 3.2 Open and Closed Sets

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  • Опубліковано 23 гру 2024

КОМЕНТАРІ • 8

  • @valeriereid2337
    @valeriereid2337 2 роки тому +1

    Excellent lecture, this is why I love mathematics. Thank you again for another great upload.

  • @emergencysocks
    @emergencysocks Рік тому

    Why is R and the empty set, ø, both closed and open sets? I thought R and the empty sets were only open-never closed. I see it listed twice as an example for the open set (at 8:58) and the closed set (22:03).

    • @MarcRenault
      @MarcRenault  Рік тому +1

      Right: R and ø are both open and closed. I know it's a little weird, and properties of the empty set in particular can be nonintuitive. The idea is that by the *definition* of open and closed sets, you can convince yourself that R is both open and closed. Then, use Theorem 3.2.13 on complements (27:05) to deduce that ø is both open and closed.

    • @emergencysocks
      @emergencysocks Рік тому

      @@MarcRenault Thank you!

  • @thnxm8
    @thnxm8 4 місяці тому

    thanks

  • @algorithmo134
    @algorithmo134 3 роки тому

    Do you have videos on Analysis II implicit function theorem, contraction mapping, inverse function and so on?

  • @jiahao2709
    @jiahao2709 Рік тому

    in the forward proof, how to make sure each a_n are different from the 1/n neigborhood of x?

  • @algorithmo134
    @algorithmo134 3 роки тому

    sir do u have videos on real analysis ii