Exercise 3.5.10

Question

[Baire's Theorem]
  Prove set of real numbers $\mathbf{R}$ cannot be written as the countable union of nowhere-dense sets.

To start, assume that $E_{1}, E_{2}, E_{3}, \ldots$ are each nowhere-dense and satisfy $\mathbf{R}=\bigcup_{n=1}^{\infty} E_{n}$ then find a contradiction to the results in this section.

Ulisse Mini · Accepted Answer

By the definition of $E_n$ being nowhere-dense, the closure $\overline{E_n}$ contains no nonempty open intervals meaning we can apply Exercise 3.5.5 to conclude that
  $$
  \bigcup_{n=1}^\infty \overline{E_n} 
e \mathbf{R}
  $$
  Since each $E_n \subseteq \overline{E_n}$ we have
  $$
  \bigcup_{n=1}^\infty E_n 
e \mathbf{R}
  $$
  as desired.

Exercise 3.5.10

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Exercise 3.5.10

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