As the title suggests, I have to prove, if (X, d) is a metric space, that for any subset A of X, diam A = diam Cl(A), i.e. the diameter of A equals the diameter of its closure.(adsbygoogle = window.adsbygoogle || []).push({});

So, if A is closed, it is trivial, since Cl(A) = A. Assume A is open. Now I'm a bit lost.

If A is open in (X, d), then it is a union of open balls in X. By the way, I know that for every open ball K(x, r) in X, diam K = diam Cl(K). Also, I have shown that Cl(A) can be written as a union of closed sets in X (since the closure of a union equals the union of closures).

Also, I know that, since A is a subset of Cl(A), diam A <= diam Cl(A) must hold.

And also, I know that for subsets A, B of X, diam (A U B) <= diam A + diam B + d(A, B), which can be generalized for any finite number of subsets.

The definitions of diameters didn't get me anywhere, too.

For some reason, I can't make any use of these facts to prove it. Any suggestion?

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# Diam A = diam Cl(A) in metric space

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