Understand The Baire Category Theorem: Dense Sets, Nowhere Dense Sets, & Infinity

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Need help understanding dense sets, nowhere dense sets, and the Baire Category Theorem? Want another way to prove that the real number are uncountable? Here's a detailed treatment of these topics. Thanks for watching, check out the rest of the playlist!
  1. Andrew McCrady
  2. Baire Category Theorem
  3. Dense set
  4. Nowhere Dense Set
  5. uncountable
  6. complete metric space
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Комментарии
Автор

So far the most intuitive explanation of the concept of nowhere dense sets that I saw on the internet. Thank you for making this video!

shramini
Автор

Incredible explanation. Thanks so much!!!

jakegameroff
Автор

I see. You’re right that I’m not using the definition in 2:05. Instead I’m showing an equivalent statement that R/Z is a dense open subset of R.

DrMcCrady
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In 8:40 you say we're allowed to take that radius r_1 such that 0<r_1<1. Is it so because as B(x, r) \cap U_1 is open then we can find a radius l>0 where B(y_1, l) is contained in B(x, r) \cap U_1 but if we take a real number r_1 such that 0<r_1<min(1, L) small enough (the density of the rationals or the Archimedean property can help us control its smallness) such that the closure of B(y_1, r_1) (the ball itself with the point z in X such that d(z, y_1) = r_1 ) is contained in B(x, r) \cap U_1? And is it the same reasoning for 9:26, right?

jeanveramorocho
Автор

2: 15 I do not understand how the example is tied to the definition. Why do we take x outside Z ? Based on the definition, shouldn't we take x from the closure of Z which is Z again??

Tatiana-zsdc