Why would we want an immersion? Immersion:   A mapping from a smooth manfold into another one is said to be an immersion if at every point the derivative mapping is an injective mapping of the first manifold's tangent space to the second one. Whitney Immersion Theorem:  Any smooth n-dimensional manifold can be immersed in R2n-1 Another was to think of an immersion: Given any point of the first manifold, I can find a small enough neighborhood so it's image in the second manifold (often a Euclidean space) is the topological equivalent of a disc (or open ball). - Immersion may self intersect, but at least locally it doesn't "look like" it.
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