Complete metric spaces are Baire spaces- a discussion of the proof.

I refer to the proof of the statement “Every complete metric space is a Baire space.”
The proof of this statement, as given in “Introduction to Banach spaces and their Geometry”, by Bernard Beauzamy, is

Let $U_1,U_2,\dots$ be a countable set of _open_ dense subsets of complete metric space $X$ . Take any open set $W$ . We will prove that $(\bigcap U_i)\cap W$ is non-empty. $U_1\cap W\neq\emptyset$ . For point $x_1\in U_1\cap W$ , construct, $\overline{B}(x_1,r_1)\subset U_1\cap W$ . Here $\overline{B}(x_1,r_1)$ is the closure of $B(x_1,r_1)$ . Not take the intersection of $U_2$ and $\overline{B}(x_1,r_1)$ , and construct $\overline{B}(x_2,r_2)\subset U_2\cap \overline{B}(x_1,r_1)$ . Similarly, for every $n\in\Bbb{N}$ , construct $\overline{B}(x_n,r_n)\subset U_n\cap \overline{B}(x_{n-1},r_{n-1})$ . Also, ensure that $0<r_k<\frac{1}{k}$ ; in other words ensure that $r_k$ converges to $0$ .

It is clear that the points $\{x_1,x_2,\dots\}$ form a cauchy sequence. As $X$ is complete, the limit $l$ of this point exists in $X$ . This point is present in all $U_n$ , and also in all $\overline{B}(x_n,r_n)$ (as a result, it is present in $W$ ). Hence, it is present in $\bigcap U_i$ . Hence $(\bigcap U_i)\cap W$ is not empty. This proves the theorem.

We have taken one open set $W$ , and proved, $(\bigcap U_i)\cap W$ is not empty. Should we take another open set $T$ , we will again be able to prove $(\bigcap U_i)\cap T$ is not empty. Note that this does not in any way imply that $(\bigcap U_i)\cap W=(\bigcap U_i)\cap T$ . All that it says is that any open set $W$ has a non-empty intersection with $(\bigcap U_i)$ , which is sufficient for $(\bigcap U_i)$ to be a dense set in $X$ .

Why do we not take $B(x_k,r_k)$ instead of $\overline{B}(x_k,r_k)$ ? This is because if the balls are not closed, then it would be difficult to prove the limit point of the sequence $\{x_1,x_2,\dots\}$ is also present in $\bigcap_i B(x_i,r_i)$ . Try this for yourself. Also, it is ok to take closed balls because if $U_i$ has a non-empty intersection with $B(x_{i-1},r_{i-1})$ , then it definitely has a non-empty intersection with $\overline{B}(x_{i-1},r_{i-1})$ , and we are only concerned with getting an intersection so that we can construct another closed ball inside it. It is elementary to prove that the limit point of $\{x_1,x_2,\dots\}$ has to lie inside the intersection of all the closed balls (hint: $\overline{B}(x_k,r_k)$ contains all $x_j$ of the cauchy sequence $\{x_1,x_2,x_3,\dots\}$ , where $j\geq k$ ).

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Hello! My name is Ayush Khaitan, and I'm a graduate student in Mathematics. I am always excited about talking to people about their research. Please please set up a meeting with me if you feel that I might have an interesting perspective to offer- https://calendly.com/ayushkhaitan/meeting-with-ayush View more posts

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