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Math Help - Embeddings

  1. #1
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    Embeddings

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

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

    For the sphere \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 \mathbb{Re}^n. That implies that there is an embedding of 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

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