How to solve this?
Let $f\colon M \to N$ and $g\colon N\to L$ be smooth maps.
- Show that if $f$ and $g$ are embeddings, then $g\circ f$ is an embedding.
- Show that if $g$ and $g\circ f$ are embeddings, then $f$ is an embedding.
- Find an example where $f$ and $g\circ f$ are embeddings but $g$ is not an embedding. (You can find such an example where $M$, $N$, and $L$ are open subsets of $\mathbb{R}$).
Edit:
Can only use this definition of an embedding:
A smooth map f:N->M is an embedding if
(I) It is a one-to-one immersion and
(II) the image f(N) with the subspace topology is homeomorphic to N under f.
I think I have shown (I), for 1. and 2., how to show (II)?