Suppose we have a metric space $(X,d)$. I wonder whether finding the closure of it is the completion of the metric space?
My thinking
I do know the defination of the completion of a metric space. It seems it's more general and complex than just find the closure. But why we need such a complex defination if clusure can make the space complete?
Consider $(0,1) \cap \mathbb{Q}$. Its closure in $\mathbb{Q}$ is $[0,1] \cap \mathbb{Q}$, but this space is not complete. Its completion would be $[0,1]$, as a subset of real numbers.
Similarly, $\mathbb{Q}$, $(0,1)$, an open ball in any metric space, etc. are closed in themselves, but are not equal to their completion. Recall that a space $Y \subset X$ is closed in $X$ if the complement $X \backslash Y$ is open. If $Y=X$, then this complement is the empty set, and the empty set is certainly open.