I am trying to prove that the Whitney sum of the Mobius bundle is trivial. I have read here: Global Trivialization of $M\oplus M$ a way to do it, but I don't know nothing about sections.
I am trying to prove it by defining a vector bundle isomorphism between $M\oplus M$ and $M\times \mathbb{R}^2$.
The elements of $M \oplus M$ are of the form $(e^{2\pi ti},([y_{1},y_{2}],[z_{1},z_{2}]))$ where $t\in [0,1]$ and $([y_{1},y_{2}],[z_{1},z_{2}])\in M\times M$ with $y_{1}=y_{2}=t$ if $t\neq 0,1$ and $y_{1},y_{1}\in \{0,1\}$ if $t\in \{0,1\}$. Hence, if $x\neq (1,0)$, I can send that point to $(x,(y_{1},y_{2})$), but if $x=(1,0)$ I don't know how to define the map in order to be well-defined and continuous.
Can anyone help me, please? Thanks in advance.