Is the space $\mathbb{N}$ is a baire spaces ? yes/No

Question

Is the space $\mathbb{N}$ is a baire spaces ?

My attempts : No , It will not form a baires spaces ,

let $C_n= \mathbb{N}\setminus\{n\}$. This is open (since $\{n\}$ is closed in $\mathbb{R}$, hence in the induced topology of $\mathbb{N}$) and dense, since every open ball with center in $\mathbb{n}$ intersects ${C_n}$. But $$\bigcap_{n\in\mathbb{N}} {C}_n = \emptyset$$

So $\mathbb{N}$ is not a baires spaces because for Baire Space the intersection of a countable family of open dense sets must be dense.

Is Its correct or not ?

Answer 1

Yes, $\mathbb{N}$ is a Baire space, assuming $\mathbb{N}$ is equipped with the usual topology. The reason is the Baire Category theorem; $\mathbb{N}$ is a complete metric space (all Cauchy sequences are eventually constant!), and hence is a Baire space.

The reason your disproof doesn't work is because $C_n$ is not dense. In particular, $C_n \cap B(n, 1/2) = \emptyset$; there's no natural numbers other than $n$ within a distance of $1/2$ from $n$. That is, $n \notin \overline{C_n}$.