Is this Proof on Manifold Correct?

Question

I want to prove this

Prove that the set of vectors $x\in \mathbb{R}^n$ with pairwise distinct coordinates is a manifold. Hence, find its dimension.

Attempt

Let $M$ be the set of vectors in $\mathbb{R}^n$ with pairwise distinct coordinates. We know $\mathbb{R}^n$ is a manifold, and $M\subseteq \mathbb{R}^n$. All we have to do is prove that $M$ is an open set (since an open subset of a manifold is a manifold).

To prove that $M$ is an open set, for any $x=(x_1,…,x_n)\in M$, consider $B_\epsilon (x) = \{(y_1,…,y_n)|\sqrt{(y_1-x_1)^2 +…+ (y_n-x_n)^2} <\epsilon \}$, where $\epsilon >0$. We can choose $\epsilon$ so small that the coordinates of any vector in this ball will be sufficiently close to the corresponding coordinates of $x$. Therefore, no two distinct coordinates can be equal. So, $B_\epsilon (x)$ is entirely contained in $M$. Hence, $M$ is open. And its dimension is n.

Question

I feel my prove of $M$ being an open set is weak. Is there something I am getting wrong here? Is there a way I can improve on this proof?

Again, is $n$ the dimension of $M$?

Your help would be really appreciated.

Note: I tried to put brace brackets in the definition for the open ball by using “\ {“ and “}\” but the brace brackets didn’t display.

Answer 1

Your idea is absolutely correct. Your strategy is to prove that $M$ is open in $\mathbb R^n$ which implies that $M$ is an $n$-dimensional manifold. However, it should be elaborated a bit more precisely.

Let $x \in M$. Define $$r = \min \{ \lvert x_i - x_j \rvert \mid i,j \in \{1,\ldots,n \} \text{ with } i \ne j \} .$$ This is a positive number. We claim that $B_{r/2}(x) \subset M$.

So let $y \in B_{r/2}(x)$ which means $\sqrt{\sum_{i=1}^n (y_i - x_i)^ 2} < r/2$. Hence all $\lvert y_i - x_i \rvert = \sqrt{(y_i - x_i)^ 2} < r/2$. Assume that $y_i = y_j$ for some $i,j \in \{1,\ldots,n \}$ with $i \ne j$. Then $$\lvert x_i - x_j \rvert = \lvert (x_i - y_i) + (y_i - y_j) + (y_j - x_j) \rvert = \lvert (x_i - y_i) + (y_j - x_j) \rvert \le \lvert x_i - y_i \rvert + \lvert y_j - x_j \rvert \\ < r/2 + r/2 = r$$ which is a contradiction.