Non-deterministic TM and problems in class P Definition:
$P$ is the set of languages that can be decided by a deterministic Turing Machine in polynomial time.

I wonder if:
$P$ equals to the set of languages that can be decided by a non-deterministic Turing Machine in polynomial time?

I think this also has to be true in this case because a non-deterministic Turing Machine can also decide all problems which can be decided by a deterministic Turing Machine. Because of the fact that we have here a language in P this NTM also decides it in P.

Am I right or am I wrong?
The class $P$ is a subset of $NP$. For an $NP$ algorithm to simulate $P$ in polynomial time, it does not even need to use nondeterminism, since a single branch can solve the $P$ problem. 

It could be a bit confusing though when textbooks say that an $NP$ algorithm works in 'polynomial time'. But this refers to the amount of time it takes for a single branch to compute. But taken as a whole, $NP= \cup NTIME(n^k)$, while $P= \cup DTIME(n^k)$.