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).
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.
Excellent lecture, this is why I love mathematics. Thank you again for another great upload.
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).
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.
@@MarcRenault Thank you!
thanks
Do you have videos on Analysis II implicit function theorem, contraction mapping, inverse function and so on?
in the forward proof, how to make sure each a_n are different from the 1/n neigborhood of x?
sir do u have videos on real analysis ii