Immersed submanifold

An immersed submanifold of $M$ is a manifold $N$ with an injective immersion $\iota :N\to M$; when $N$ is treated as a subset, its manifold topology can be finer than its subspace topology.

For irrational $\alpha$, the mapis an injective immersion with dense image. It is not a smooth embedding, since points with arbitrarily large parameter return arbitrarily close to its initial point in the subspace topology.

An injective immersion from a compact smooth manifold into a Hausdorff smooth manifold is a smooth embedding. Indeed, it is a continuous bijection onto its image, and compactness makes its inverse continuous.