TITLE: what is the difference between a set's closure and completion?
QUESTION [12 upvotes]: I know that when we talk about completion of a space, we infer that it is a metric space. So can I say that: Given a complete metric space $M$ and a subspace of $M$, $S$, the closure of $S$ is a completion of $S$?

REPLY [5 votes]: No. The closure of a set happens inside a given space $X$. Completion on the other may require adding many new points also to the ambient space. For instance, take $X=\mathbb Q$ with its standard metric. The closure of $\mathbb Q$ is $\mathbb Q$, but the completion is $\mathbb R$. 
Later edit, following adding the 'complete' to the question, the answer is 'yes'.