In general the closure of the ball B(a, r) need not be the closed ball B_c(a, r). Take for example a set containing more than one point, and define the metric d by d(x, y) =O iff x=y and d(x,y) =1 otherwise. Then the open ball B(a, 1) is reduced to {a} which is closed, but the closed ball B_c(a, 1) is the whole space. However the closure of B(a, r) may be a closed ball with a different radius. In this example {a} =B_c(a, 1/2).
In general the closure of the ball B(a, r) need not be the closed ball B_c(a, r). Take for example a set containing more than one point, and define the metric d by d(x, y) =O iff x=y and d(x,y) =1 otherwise. Then the open ball B(a, 1) is reduced to {a} which is closed, but the closed ball B_c(a, 1) is the whole space. However the closure of B(a, r) may be a closed ball with a different radius. In this example {a} =B_c(a, 1/2).
That is may be hold for some special type of metric space, try to find the condition that make it happen.
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