In the definition of (how to construct) a tangent vector with respect to a given manifold $M$, at a particular point $x \in M$, described here as equivalence class of certain curves through $x$, it is explicitly noted (where I added emphasis to the quote):
The tangent space of $M$ at $x$ [...] is defined as the set of all tangent vectors [at $x$];
it does not depend on the choice of chart [...]
I'd like to understand whether and why such independence holds specificly for the following example choices of charts of a suitable open subset $U$ of a given 2-dimensional manifold $M$:
Consider
- $\varphi : U \longleftrightarrow \{ \, (a, b) \in \mathbb R^2 \, | \, (-1 \lt a \lt 1) \text{ and } (a - 1 \lt b \lt 1 - a) \, \}$
explicitly being a homeomorphism from $U$ (wrt. its topology given by 2-dimensional manifold $M$ as a topological space) to the described open subset of $\mathbb R^2$ (wrt. its usual topology);
and having an inverse, $\varphi^{-1}$, which is a homeomorphism as well);
therefore $(U, \varphi)$ explicitly being a chart, where explicitly $x \equiv \varphi^{-1}[ \, (0, 0) \, ] \in U$,
and
- function $f : \mathbb R^2 \longleftrightarrow \mathbb R^2, \qquad f[ \, (a, b) \, ] \mapsto (p, q) \equiv (a + $$\text{Abs}$$[ \, b \, ] ,b) $.
My questions:
(1) Is it correct that function $f$ is a homeomorphism (wrt. usual topology) ?
(2) Is it correct that $\psi : U \longleftrightarrow \{ \, (p, q) \in \mathbb R^2 \, | \, (-1 \lt p \lt 1) \text{ and } (-1 - p \lt q \lt 1 + p) \, \}, \qquad \psi \equiv f \circ \varphi, $
is a homeomorphism as well; and that consequently $(U, \psi)$ is a chart, too ?
Further, returning to the definition of tangent vectors referenced above:
(3) Does the curve $\gamma : (-1, 1) \to M, \qquad \gamma[ \, t \, ] \mapsto \varphi^{-1}[ \, (0, t) \, ],$
belong to an equivalence class of curves which constitutes a tangent vector on $M$ in point $x \equiv \varphi^{-1}[ \, (0, 0) \, ]$ ?,
and
(4) Does the curve $\rho : (-1, 1) \to M, \qquad \rho[ \, \tau \, ] \mapsto \psi^{-1}[ \, (0, \tau) \, ],$
belong to an equivalence class of curves which constitutes a tangent vector on $M$ in point $x \equiv \psi^{-1}[ \, (0, 0) \, ] = \varphi^{-1}[ \, (0, 0) \, ]$ ?
Here's an illustration of the two curves $\gamma$ and $\rho$, projected into the ranges of chart $(U, \varphi)$, and of chart $(U, \psi)$, respectively:
