An immersed submanifold is by definition the image of a smooth immersion. I know some examples but I lack general understanding of what immersed submanifolds look like. For example, can one characterize subsets of a manifold $M$ that are immersed submanifolds of a given dimension?
For example, subsets of $M^n$ that are embedded smooth $k$-dimensional submanifolds $S$ are those for which $(M, S)$ is locally diffeomorphic to $(\mathbb R^n, \mathbb R^k)$, so by looking at $S$ one can instantly tell if it is a smooth submanifold. How does one do the same for immersed submanifolds? Is the union of countably many embedded submanifolds an immersed one? Are there any immersed submanifolds that cannot be decomposed as the union of countably many embedded submanifolds?
