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.
Thus, if we add those inequalities, we have
Then, choose for .
Hence, must be open.
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.