(1) Is $U=\{(x,y)\in \mathbb{R^2} : x^2+y^2 \neq 1\}$ open in $\mathbb{R^2}$?
(2) How can i show $(a,b)\times(c,d)$ is open in $\mathbb{R^2}$?
(3)Is $S=\{(x,y)\in \mathbb{R^2}:xy\neq 0\}$ open in $\mathbb{R^2}$?
In all these examples i draw the figure in (1) it is interior part of circle and exterior part of circle so i can draw any ball with some radius it is also containing in the set.
Similar argument for (2) which is rectangle region in $\mathbb {R^2}$ and in the (3) is unbounded region without containing X and Y axis. It is very clear from the geometry but Anyone help me how can i prove all these three sets are open using definition of open set or triangle inequality? If it is possible then give me some hints. Thanks in advance.