I'm reading John B. Conway's book on point-set topology and there the Baire category theorem is stated as follows:
If $(X,d)$ is a complete metric space and $\{U_n\}$ is a sequence of open subsets of $X$ each of which is dense, then $\bigcap_{n=1}^\infty U_n$ is dense.
Does this mean that the subsets are dense in $X$ or in some subset of
$X$ and why is their intersection always nonempty? For example
$\mathbb Q\cap\mathbb R\setminus\{\mathbb Q\}=\emptyset$ and they are
both dense in $\mathbb R$?
Of course this is not an infinite sequence
of subsets but to me it feels like pointing to the direction that the
intersection might be empty. The proof uses the fact that the
intersection is nonempty.
Best Answer
As noted in comments about, $\mathbb Q$ and $\mathbb R\setminus \mathbb Q$ are dense, but they are not open.
It is easier to characterize the complement $V$ of open dense subset of $\mathbb R.$
$V\subseteq\mathbb R$ is closed and contains no open interval if and only if $\mathbb R\setminus V$ is open and dense.
Examples of such $V:$
Such sets tend to be “small” in comparison with the whole set. These examples are all measure zero, for example. The general case might not have a measure, but the theorem essentially gives another way of saying such sets are “small.”
Another way to state the theorem is if $\{V_i\}$ is a countable set of closed sets, none of which contains an open ball, the their union does not contain an open ball.