Let (X, d) be a metric space and let a ∈ X.
How would you read this statement ? $(B_r(a))'$ $\subset$ $\,$$B_r[a]$
What I think this statement says is "not" the open ball with center a and radius r is a subset of the closed ball with center a and radius r.
I have to prove this statement but I am not sure if I completely understand what the statement means.
A hint on how to start the proof would be much appreciated too as I am stuck on that as well.
The complement of the open ball might not necessarily be contained in the closed ball. Take the open ball of radius 1, centered at 0 in $\mathbb{R}^2$. Clearly it's complement is not contained in the closed ball, which is just the open ball, with its boundary. However, it is true than every open ball is contained within the closed ball. On the other hand, the punctured open ball is contained within the open ball and thus within the closed ball.