Graph Parametrization of Continuous Mapping is Embedding

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Theorem

Let $U \subseteq \R^n$ be an open subset of $n$-dimensional Euclidean space.

Let $f : U \to \R^k$ be a continuous mapping.

Let $\map \Gamma f$ be the graph of $f$.

Let $\gamma_f : U \to \R^n \times \R^k$ be the graph parametrization of $\map \Gamma f$.

Then $\gamma_f$ is an embedding of $\R^n$ to $\R^{n + k}$.

Proof

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Sources

• 2011: John M. Lee: Introduction to Topological Manifolds (2nd ed.) ... (previous) ... (next): $\S 3$: New Spaces From Old: Subspaces. Topological Embeddings