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I read a Lemma Introduction to smooth manifolds" which says that given a smooth n -manifold 𝑀, then the tangent bundle 𝑇𝑀 is a smooth 2𝑛-manifold, how can I prove this?

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    $\begingroup$ I assume you're familiar with how charts are chosen in $T(M)$, but you want to know why it is actually Hausdorff? This is what I got from the title, but it's not in the body of the question. $\endgroup$ – D_S Jun 24 '19 at 15:51
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We need to study the 2 possibles cases:

Considering $\Pi$, the application, such that:

$\Pi: E \rightarrow M$

If:

$\Pi(x) \neq \Pi(y)$

Since $M$ is Hausdorff there are $U,V$ disjoint, such that: $ \Pi (x) \in U $, $ \Pi (y) \in V$

Then $x \in \Pi ^{- 1}(U)$ and $y \in \Pi^{- 1}(V)$,

Also, $U,V$ are disjoint

$ U \cap V \neq \emptyset $

$ \Pi ^{- 1} (U) \cap \Pi^{- 1} (V) \neq \emptyset. $

If:

$\Pi(x) = \Pi(y) = p$.

By definition for each point $p$, there exists a $p \in U $ neighbourhood of the point that preserves a fiber differencing:

$ \theta: \Pi^{- 1} (U) \rightarrow U \times R^{ n} $.

Then we have:

$\theta(x) = (p, x^{'})$ $ \theta(y) = (p, y^{'})$

Where $ x^{'}, y^{'} $ are vectors in $ R^{n}$

Since $\theta $ is a diffeormorphism, $ x^{'}\neq y^{'} $.

Also $ R^{n}$ is Hausdorff.

The neighbourhoods $ H, U \in R^{n}$ such that $ x^{'}\in H, y^{'} \in V $

Therefore,

$(p, x^{'}) \in U \times H$

and

$(p, y^{'}) \in U \times V$.

Where $ x \in \theta^{- 1} (U \times H)$ and $ y \in \theta^{- 1} (U \times V) $ are disjoint environments.

We concluded that the tangent bundle $TM$ of a manifold $M $ is Hausdorff.

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Let $v$ and $w$ be two distinct points in $T(M)$. Say that $v \in T_x(M)$ and $w \in T_y(M)$ for some $x, y \in M$. If $x = y$, then $v$ and $w$ can definitely be separated by an open set in $T(M)$, since locally $v$ and $w$ are given by the last $n$-coordinates in $\mathbb R^n \times \mathbb R^n$.

If $x \neq y$, choose disjoint charts $U$ and $V$ containing $x$ and $y$, respectively. We can do this because $M$ is Hausdorff. Then

$$\bigcup\limits_{p \in U} T_p(M) \cong U \times \mathbb R^n$$

$$\bigcup\limits_{p \in V} T_p(M) \cong V \times \mathbb R^n$$

are disjoint open sets of $T(M)$ containing $v$ and $w$, respectively.

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