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I know the properties of the metric:

d(x,y) > 0 if x != y, d(x,x) = 0

d(x,y) = d(y,x)

d(x,y) + d(y,z) >= d(x,z)

Now take any open set (a,b) in R (im assuming the standard topology on R). d^-1((a,b)) = {(x,y) e X x X : d(x,y) e (a,b)} (e stands for element)

Now i have to show d^-1((a,b)) is open. I tried playing around with the properties in different cases, but i dont have a clear indication of how to move on from here. I will keep trying but if anyone can guide me that would be good.