Baire property of complete metric spaces Maybe it's a silly question but I would like to have some heuristic (if it's possible) about the Baire property for complete metric spaces.
The property that I'm trying to understand is stated as follows:
"A metric space $(X,d)$ is said to have the Baire property if the intersection of any sequence of dense open sets of $X$ is dense in X, i.e.:
For any $(U_n)_n$ open subset of $X$ with $\overline{U_n}=X$ for any $n \in \mathbb{N}$ it holds
$$\overline{\bigcap_n U_n}=X$$"
I would like to know if it's possible to interpret this property in some way. An example of metric space with such a property would be nice too.
Thanks in advance
 A: In $\mathbb{R}$ dense open sets will be of the form $\mathbb{R}-C$, where $C$ is any countable set which is nowhere dense. For example, $\mathbb{R}-\mathbb{Z}$ is open and dense in $\mathbb{R}$. Now see the intersection of such open dense sets in $\mathbb{R}$.
A: I don't know if this answers your question but its something that you need to know when it comes to Baire spaces.
Baire's theorem has this useful corollary.

Let $(X,d)$ be a complete metric space and $X= \bigcup_{n=1}^{\infty}F_n$ where $F_n$ are closed subsets of $X$.Then at least on of these closed sets has a nonempty interior.

This property is used more often than the original statement of the theorem,especially in functional analysis.
This corollary says that a complete metric space cannot be expresed as a union of topologicaly small sets which are the nowhere dense sets.
Now with this property,for instance you can see(which is intuitively obvious) that $(\mathbb{R}^2,d_2)$  cannot be expressed as a countable union of lines.
$(\mathbb{R}^2,d_2)$ is a complete metric space with respect to $d_2$(the euclideian metric) but the conlusion fails because a line is a closed set with empty interior,in other words a nowhere dense subset.
From Baire's theorem you can deduce that a complete metric space is a space of $second-category$.
A space of $first-category$ is a space $X$ that can be expressed as a countable union of nowhere dense subsets.
A space of $second-category$ is a spcace that is not of $first-category$.
Now from this corollary you have another proof that $\mathbb{R}$ is uncountable.
If $\mathbb{R}$ was countable then it would be expresed as a countable union of singletons.But $\mathbb{R}$ is a complete metric space with respect to the euclideian metric and every singleton has an empty interior.Therefore we contradict the Corollary of Baire's theorem.
Now as an exercise you can prove from Baire's theorem the Corollary.
I hope this helps a little to understand this great theorem,but i believe it does not answer fully your question.
A: The intersection of dense sets need not be dense: consider the rationals and the irrationals in the reals, both of which are dense subsets.
But it is quite easy to see that the intersection of finitely many open and dense sets is dense, in any space. So it's quite natural to ask for spaces where this holds for countably many open and dense sets. This does not always hold, because in $X = \mathbb{Q}$ all sets of the form $X \setminus \{q\}$ are open and dense, but $\bigcap \{X \setminus \{q\} : q \in \mathbb{Q}\} = \emptyset$, while in a discrete space $X$ (like $\mathbb{Z}$ in the usual topology) the only open and dense set is $X$, so this trivially holds.
But in the reals, complements of points are open and dense, and if we look at 
$\bigcap \{X \setminus \{q\} : q \in \mathbb{Q}\}$ there, this is just the irrationals (which are dense). It turns out that complete metric spaces and locally compact Hausdorff spaces are both classes of spaces where Baireness holds. But there are also Baire spaces which are not of this type, e.g. $\mathbb{R}^I$ where $I$ is uncountable. But even metric spaces that are Baire need not be complete. But is's in a way "close to being complete". All spaces $\mathbb{R}^n$ have the Baire property, as do the space of irrationals as a subspace of the reals. Most spaces occurring in functional analysis do. We can prove theorems like the open mapping theorem and the closed graph-theorem and the uniform boundedness principle for them (which is why Banach spaces, which have the Baire property, are more useful than just plain normed spaces.
