I'm quite new to topology. Just wondering if anyone can help me out a bit.

QuestionLet Show is an open subset of

My Attempt

Let .

We want to show that there exists an such that .

Suppose that .

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.