The definition of a submanifold

Question

I am wondering why it is insufficient to define a submanifold of a manifold $M$ as a subset $S\subset M$ such that $S$ itself is a manifold. Why do we need the notions of embedded submanifolds or immersed submanifolds?

Answer 1

It's for the same reason we require that a subgroup $H$ of $G$ (for example) has the same operation product than the one in $G$, or that a linear subspace has the same addition and scalar scaling than the whole space, etc. Because otherwise you're just talking about an abstract set $S$ that simply happen to be contained in the manifold $M$, and all that matters is its cardinality.

Here's a silly example. Take $M = \mathbb{R}^2$ and take $S = (\mathbb{R} \times \{0\}) \cup \mathbb{Q}^2$. It's obviously not a submanifold of $M$ in any possible meaning of the word, but it has the same cardinality as $\mathbb{R}^{18}$ and by transferring the structure of $\mathbb{R}^{18}$ through the bijection, you could make it into an $18$-dimensional manifold...