Why is a line a closed subset of $\mathbb R^2$? I'm studying topology and I have a doubt in the following exercise. I'd appreciate some help. Let $m\neq0$ and $c$ be real numbers. Prove that the line $L=\{\langle x,y\rangle:y=mx+c\}$ is a closed subset of $\mathbb R^2$. I found similar questions here, but with answers involving continuity of functions. This exercise is in chapter $2$ of Topology without tears, in which continuity is presented in the fifth. Then, my attempt was to prove that $\mathbb R^2 \backslash L$ is open by setting, at each point $p\in L$, two open rectangles with a vertex at $p$: one above $L$ and the other below. As $\mathbb R^2 \backslash L$ is the union of these rectangles $(?)$, and every one of them is a open subset of $\mathbb R^2$, we have done. My guess is right? Thank you!

If you know that the real line is a closed subspace of $\Bbb{R}^2$ (indeed, it's a complete metric subspace, which is even stronger) the result follows easily because the real line can be mapped to any other line in $\Bbb{R}^2$ by a composition of a rotation and a translation. Since rotations and translations are isometries (i.e. homeomorphisms) of $\Bbb{R}^2$, they preserve closed sets.
