The topology of $T_pM$ [closed]

Let $$M$$ be smooth $$n$$-manifold with Riemannian metric $$g$$. If I consider the tangent bundle $$TM$$, then I can see that the tangent space at a point $$p\in M$$ is an embedded submanifold of $$TM$$. Indeed, called $$\pi\colon TM\longrightarrow M$$ the canonical projection, and taken a chart $$(U,\phi)$$ of $$M$$, the chart $$\pi^{-1}(U)=TU\longrightarrow \phi(U)\times \mathbb{R}^n$$ gives us a diffeomorphism between the embedded submanifold $$\phi(p)\times\mathbb{R^n}\subseteq \mathbb{R}^{2n}$$ and $$T_pM$$. I would prove that the topology of $$T_pM$$ induced by the Riemannian metric $$g$$ is the same of the topology of embedded submanifold of $$TM$$.

In this way, if the claim above is true, I obtain that the set $$V=\lbrace v\in T_pM \: : \: |v|<\epsilon\rbrace$$ (for some $$\epsilon >0$$), where we define the exponential map $$\exp_p\colon V\longrightarrow M$$, is open.

closed as unclear what you're asking by Moishe Kohan, José Carlos Santos, kingW3, jvdhooft, Lord Shark the UnknownJul 2 '17 at 13:42

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• What exactly is your question? – Math1000 Jul 2 '17 at 0:24