No! you can't talk about such a Buntilyou have shown that it exists!

Better: if then it is NOT on the line x+ y= 0. Therefore, the distance from to that line is not 0. Let be that distance and show that no point in the neighborhood of with radius is on that line.

Then, .

So, .

Thus, if we add those inequalities, we have

Then, choose for .

Thus, and

Hence, must be open.

Comment

Is the above working correct?

Is using the infinity distance function instead of the euclidean distance function, mathematically correct? Could I have used something like instead?

Is there anyway to simplify the working out? What is the best way choosing an epsilon?

Thanks in advance.