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Question Number 165756 by CrispyXYZ last updated on 07/Feb/22

Let M be a compact smooth manifold of dimension d.  Prove that there exists some n ∈Z^+  such that M can be regularly embedded in the Euclidean space R^n .

$$\mathrm{Let}\:{M}\:\mathrm{be}\:\mathrm{a}\:\mathrm{compact}\:\mathrm{smooth}\:\mathrm{manifold}\:\mathrm{of}\:\mathrm{dimension}\:{d}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{there}\:\mathrm{exists}\:\mathrm{some}\:{n}\:\in\mathbb{Z}^{+} \:\mathrm{such}\:\mathrm{that}\:{M}\:\mathrm{can}\:\mathrm{be}\:\mathrm{regularly}\:\mathrm{embedded}\:\mathrm{in}\:\mathrm{the}\:\mathrm{Euclidean}\:\mathrm{space}\:\mathbb{R}^{{n}} . \\ $$

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