As a second question: can we guarantee the completness of (the subset as subspace) a open subset $L$ of a complete metric space $M$ ?
Complete subspace $L$, of metric space $M$, is a closed set in $M$
Closed set, of complete metric space $M$, is a complete subspace in $M$
In $\mathbb{R}$, which is a complete metric space, subspace $(0,1)$ is not complete and it is open. Then can guarantee the completness of a subspace $L$ in a complete metric space $M$ when $L$ is closed.
The author remarks the first theorem:
if $L$ is isometric to a not-open $X \subset M $, then $L$ is not complete subspace of $M$
This remark I believe is the the key to answer my question but I don't quite understand it.
A subset of a complete metric space is complete for the induced metric if and only if it is closed. It may happen that an open subset is also closed. If the space is connected, this can only happen if the subset is the whole space or empty.