Can a connected manifold with nonempty boundary be embedded into a connected manifold without boundary and with the same dimension?

Question

Let M be a connected m-dimension manifold with nonempty boundary. My question is whether there exists a connected m-dimension manifold N without boundary and countinuous map $ f: M\to N $ such that $ f $ is an embedding? Moreover, is every manifold with boundary can be viewed as a regular domain of a manifold without boundary?