Im reading chapter9 Category, Real Analysis, Carothers, 1ed, talking about discontinuous functions of metric space. Here is a proof for a theorem that R be a countable union of closed sets,: enter image description here

Baire's theorem involved is,: enter image description here

  1. What is the "entire open interval"?

  2. Is "$E_n$ contains an interval" the same as interior of $E_n$ is not empty?

  3. if $E_n$ has empty interior, that means complement of $G_n$ has empty interior, then equivalently, this will imply that $G_n$ is dense and make a contradiction here, is that right?

  4. What im still confusing about is how can we suppose if $\mathbb{R}$ = a countable union of closed sets?

  • $\begingroup$ If the interval is $(a,b)$, the entire open interval is $(a,b)$. (As opposed to "only part of the open interval".) $\endgroup$ – Andrés E. Caicedo Dec 4 '13 at 2:28
  • $\begingroup$ @AndresCaicedo: But where is (a,b)? you mean R is (-infinity, +infinity), an open interval? $\endgroup$ – Bear and bunny Dec 4 '13 at 2:35

In response to part 4 of your question, we have $$ \mathbb R = \bigcup_{n \in \mathbb Z} [-n, n] $$ which is a countable union of closed sets.

  • $\begingroup$ Ohhh, that's right. Thank u^_^ $\endgroup$ – Bear and bunny Dec 4 '13 at 2:49
  • $\begingroup$ BTW, I can still use open (-n, n) instead of the closed one, is that right? $\endgroup$ – Bear and bunny Dec 4 '13 at 2:52
  • $\begingroup$ I mean the union of open ones will still be equal to R. $\endgroup$ – Bear and bunny Dec 4 '13 at 2:53
  • $\begingroup$ Yes, that's true. $\endgroup$ – manthanomen Dec 4 '13 at 2:55
  • $\begingroup$ @bof: cosets is what? $\endgroup$ – Bear and bunny Dec 4 '13 at 3:40

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