The set $A=\{(x, y)\in \mathbb{R}^2:|x|=|y|\}$ is connected Two disjoint sets $A$ and $B$, neither empty, are said to be mutually separated if neither contains a boundary point of the other. A set is disconnected if it is the union of separated subsets, and is called connected if it is not disconnected.

With the above definition of connected set, how to prove that the set $A=\{(x, y)\in \mathbb{R}^2:|x|=|y|\}$ is connected.

Exercise should try using the definition of previous related and for this I guess I should do the proof by contradiction but do not know how to proceed.
Do you know  straight lines are connected set in $\mathbb{R}^2$? do you know union of connected set is connected if they have one point in common?

Now can you see your set with above stated properties?
If |x| &lt; |y| then those x's are all in the disk of radius y about the origin.  That is surely connected