1. ## Embeddings

Let $\displaystyle M$ be a non-empty differentiable manifold of dimension $\displaystyle n$. Prove there exists an embedding $\displaystyle f: \mathbb{R} ^k \longrightarrow M$ for all $\displaystyle 1 \leq k \leq n$.

For the sphere $\displaystyle \mathbb{S} ^n$ it's clear that the stereographic projection is an embedding, but I can't prove the general case. Any ideas would be apreciated.

2. Originally Posted by Jose27
Let $\displaystyle M$ be a non-empty differentiable manifold of dimension $\displaystyle n$. Prove there exists an embedding $\displaystyle f: \mathbb{R} ^k \longrightarrow M$ for all $\displaystyle 1 \leq k \leq n$.

For the sphere $\displaystyle \mathbb{S} ^n$ it's clear that the stereographic projection is an embedding, but I can't prove the general case. Any ideas would be apreciated.
Isn't this applicable to all topological n-manifolds including differentiable n-manifolds?

If that is the case,

By the definition of a topological manifold, every point in n-manifold M has a neighborhood which is homeomorphic to Euclidean space $\displaystyle \mathbb{Re}^n$. That implies that there is an embedding of $\displaystyle f: \mathbb{Re}^k \longrightarrow M$ when k = n, where M is a topological n-manifold. The remaining steps are established by composing two embeddings of g and h if we define g and h such as

$\displaystyle g:\mathbb{Re}^i \longrightarrow \mathbb{Re}^n$ , $\displaystyle h:\mathbb{Re}^n \longrightarrow M$, where $\displaystyle i = 1, 2, ..., n$.