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In differential topology, the Whitney immersion theorem states that for m > 1 {\displaystyle m>1} , any smooth m {\displaystyle m} -dimensional manifold has a one-to-one immersion in Euclidean 2 m {\displaystyle 2m} -space, and a immersion in {\displaystyle } -space. Similarly, every smooth m {\displaystyle m} -dimensional manifold can be immersed in the 2 m − 1 {\displaystyle 2m-1} -dimensional sphere.
The weak version, for 2 m + 1 {\displaystyle 2m+1} , is due to transversality : two m-dimensional manifolds in R 2 m {\displaystyle \mathbf {R} ^{2m}} intersect generically in a 0-dimensional space.