In the article about metric space on wikipedia:
If $X$ is a complete subset of the metric space $M$, then $X$ is closed in $M$. Indeed, a space is complete if it is closed in any containing metric space.
Do the second sentences say that:
In a metric space, a subset is complete if and only if the subset is closed in the metric space.
If not, what is the relation between completeness and closedness for a subset in a metric space?
Thanks and regards!