# Cannot understand some parts of proof for R be a countable union of closed sets

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,:

Baire's theorem involved is,:

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?

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