" Let X be a metric space and p be a point in X and be a positive real number. Use the definition of openness to show that the set U(subset of X) given by:(adsbygoogle = window.adsbygoogle || []).push({});

U = {x∈X|d(x,p)>r} is open. "

I have tried:

U is open if every point of U be an interior point of U. x is an interior point of U if there is an open ball B(x, r) is a subset of U. Let y belongs to B(x, r) then d(x, y)< r, and p doesn’t belong to B(x, r) since d(p, x) > r. If we can show that d(y, p) > r (y is in X) then since y is a arbitrary point of B(x, r) it means that the open ball is a subset of U.

But I don’t know how to show d(y, p) > r ?

Please help me.

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# Homework Help: Open set

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