I've been trying to solve the following problem from Do Carmo's book:
Let $P = \{(x,y,z) \in \mathbb{R}^3 \mid x = y\}$ and let $\textbf{x}\colon U \subset \mathbb{R}^2 \to \mathbb{R}^3$ be given by \begin{equation} \textbf{x}(u,v) = (u + v, u + v, uv), \end{equation} where $U = \{(u,v) \in \mathbb{R}^2 \mid u > v\}$. Clearly, $\textbf{x}(U) \subset P$. Is $\textbf{x}$ a parametrization of $P$?
If this is the case, then this transformation should be a homeomorphism. So, I tried to find a topological invariant such as connectedness, and prove that the image did not satisfy the invariant. However, apparently, this transformation is a homeomorphism. Please if you can help me out with this, I will really appreciate it. Thanks in advance!