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Homework Statement
Prove that the set A = {(x,y) in R2 : xy ≠ 1} is open in the metric space (R2, d), where d is the Euclidean metric.
The attempt at a solution
A is open if for any p in A, I can find an open ball centered at p that is contained in A. This would be easy if I could find the closest point q = (x,y) to p that is not in A. Then I would just choose the open ball whose radius is d(p,q). Heck, I don't even need to find q: It suffices to know that such a q exists. This is where I'm stumped. Any tips?
Prove that the set A = {(x,y) in R2 : xy ≠ 1} is open in the metric space (R2, d), where d is the Euclidean metric.
The attempt at a solution
A is open if for any p in A, I can find an open ball centered at p that is contained in A. This would be easy if I could find the closest point q = (x,y) to p that is not in A. Then I would just choose the open ball whose radius is d(p,q). Heck, I don't even need to find q: It suffices to know that such a q exists. This is where I'm stumped. Any tips?