Proving the graph of an equivalence relation is closed in a product space Let $X = \mathbb R^n \setminus \{0\}$, $n\ge2$.
Let $\sim$ be the equivalence relation defined by $x\sim y$ iff there exists a non-zero real number $k$ such that $kx=y$.
Prove that the graph of $\sim$ in $X\times X$ is closed.
I know that I should prove that the complement of $\sim$ is open. I know that $X$ is Hausdorff. Not sure what to do next. Please help.
 A: Assume $x_n\to x$, $y_n\to y$ with $x_n\sim y_n$, say $k_nx_n=y_n$. Can you show that $k_n$ is convergent to some $k$? Then show that $kx=y$.
A: Hagen’s approach is probably the easiest, but you can do it by proving that the complement of the graph of $\sim$ is open.
Suppose that $x=\langle x_1,\dots,x_n\rangle$, $y=\langle y_1,\dots,y_n\rangle$, and $x\not\sim y$. Show that one of the following three statements must be true:


*

*There are $i,k\in\{1,\dots,n\}$ such that $x_i\ne 0\ne x_k$ and $\dfrac{y_i}{x_i}\ne\dfrac{y_k}{x_k}$.

*There are $i,k\in\{1,\dots,n\}$ such that $y_i\ne 0\ne y_k$ and $\dfrac{y_i}{x_i}\ne\dfrac{y_k}{x_k}$.

*For all $k\in\{1,\dots,n\}$, $x_ky_k=0$. Since $x,y\ne 0$, this implies that there are $i,k\in\{1,\dots,n\}$ such that $x_i\ne 0=y_i$ and $y_k\ne 0=x_k$.
In the first case use the continuity of division on its domain to show that if $x'=\langle x_1',\dots,x_n'\rangle$ and $y'=\langle y_1',\dots,y_n'\rangle$ are sufficiently close to $x$ and $y$, respectively, then $\dfrac{y_i'}{x_i'}\ne\dfrac{y_k'}{x_k'}$; this shows that there is an open nbhd about $\langle x,y\rangle\in\Bbb R^n\times\Bbb R^n$ disjoint from the graph of $\sim$. The second case is entirely similar to the first.
In the last case note that if $x'$ and $y'$ are sufficiently close to $x$ and $y$, respectively, then $y_i'$ and $x_k'$ are very near or equal to $0$, while $x_i'$ and $y_k'$ are bounded away from $0$, so $\dfrac{y_i'}{x_i'}$ is very close to $0$, and $\dfrac{y_k'}{x_k'}$ is either very large in magnitude or undefined, and in any event $x'\not\sim y'$.
I’ll leave to you the grubby details of specifying just how close sufficiently close is, should you decide to take this approach.
