Relationship between two manifolds with identical dimension I am now learning some knowledge about differential manifolds by myself. The problem I meet is that it seems that a manifold $M$ can be a proper subset of another manifold $N$ even if they have identical dimensions. But in the literature, some researchers have presented the following statement:
Given a manifold $M$ of dimension $d_m$ and a manifold $N$ of dimension $d_n$, $M\subset N$, and $p\in M$. Then $M\cap U=N\cap U$ for some neighborhood $U$ of $p$ if and only if $d_m=d_n$.
I am confused to answer that if $M$ and $N$ are two smooth manifolds with identical dimensions, and $M\subset N$
Q1. Can we conclude that $M=N$?
Q2. If the answer of Q1 is no, then can we conclude that for each $p\in M$, it holds $M\cap U=N\cap U$ for some neighborhood $U$ of $p$ ?
Q3. If the answer of Q2 is yes, how to prove the statement in Q2, is there any relevant theorems ? 
Thank you for your answer!

The answer to Q1 is straightforward: see if you can prove that any open subset of a manifold is also a manifold (of the same dimension). You just need to check that the subset is Hausdorff, second-countable, and that charts on the original manifold restrict to charts on the subset.
Q2 is a bit trickier. Pick charts $\phi$ and $\psi$ mapping to neighborhoods of $p$ on $M$ and $N$ respectively, argue that $\phi^{-1}\circ \psi$ is continuous and injective on a sufficiently small neighborhood fo $\psi^{-1}(p)$, and apply invariance of domain.
