How to prove that $\{(x,y) \in \mathbb{R}^2 \mid x \neq y\}$ is open? Prove that the subset $S := \{(x,y) \in \mathbb{R}^2 \mid x \neq y\}$ is open, with respect to a topology induced by the usual Euclidean distance.

I'm not even sure how to begin. Can anybody help me prove this?
Hint: What is its complement $S^c$? Do you know how to show that this one is closed?
HINT: Choose any point in $\mathbb{R}^2$ and show there's a bigger point using the triangle inequality for all points in $\mathbb{R}^2$.