TITLE: Cannot understand some parts of proof for R be a countable union of closed sets
QUESTION [4 upvotes]: 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,:


What is the "entire open interval"?
Is "$E_n$ contains an interval" the same as interior of $E_n$ is not empty?
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?
What im still confusing about is how can we suppose if $\mathbb{R}$ = a countable union of closed sets?

REPLY [1 votes]: 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.
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