inclusion map in smooth manifold

Question

Given smooth manifold $M$ and it's submanifold $S$(e.g. open subset of $M$) we have inclusion map $i:S\to M$.

And we treat $i$ as $i(x) = x$ typically.

For example $i:S^n \to \mathbb{R}^{n+1}$ is valid to define $i(x) = x$ But it seems not for example inclusion $i:\mathbb{R}^n \to \mathbb{R}^{n+1}$ as $(x_1,...,x_n) \to (x_1,...,x_n,0)$

So I was a bit confused what is the definition for inclusion here?Should we treat it as $i(x) = x$?

Is this "inclusion" a topological embedding by default setting or not?

I found an explanation here

Answer 1

There are actually deeper subtleties: The notion of a submanifold can breed much confusion: Do you want a submanifold to be immersed, do you want it to be an embedded submanifold?

An immersed submanifold $S$ of a manifold of $M$ is the image of a manifold under an immersion. An immersion is a smooth map with injective derivative.

An embedding is a topological embedding, i.e., a homeomorphism onto its image (with respect to the subspace topology), that is also an injective immersion.

Note!: Immersions are not necessarily injective, nor a topological embedding!

Answer 2

(1)the inclusion map is always defined as $i(x) = x$ for $i:S\to M$ such that $S\subset M$.

(2)Sometimes we may call inclusion just as a injection map,for example $i:\Bbb{R}^n\to \Bbb{R}^{n+1}$ defined above are injection.

note that both definition of "inclusion" need only set level things,no topology or smooth structure.

(For immersed/embedded submanifold defintion inclusion is the first meaning.

we call this "inclusion" the "immersion/embedding" then since there exist some additional structure on it.)