I am trying to construct the Mobius strip bundle onver $S^1$.
I am stuck in how to define charts for the Mobius bundle. Once I make this, the problem will easily be solved.
My attempt was: $$M = [0,1]\times \mathbb{R}/\sim$$ where $(0,y) \sim (1,-y)$.
Then let $U_1 = (\frac{1}{8},\frac{7}{8})\times \mathbb{R}$ an open set and $\phi_1(x,y) = (x,y)$. This is one chart for $M$.
Another one is (?):
$$U_2 = [0,\frac{1}{4})\cup (\frac{3}{4},1]\times\mathbb{R}$$ $$\phi_2(x,y) = (x,y), ~ x \in [0,\frac{1}{4}),$$ $$\phi_2(x,y) = (x,-y), ~ x\in (\frac{3}{4},1]$$
Is this correct?
If it is, then I can define the trivialization as:
$$\psi_1(x,y) = (\exp{2\pi ix},y)$$ $$\psi_2(x,y) = (\exp{2\pi ix},y), x\in [0,\frac{1}{4})$$ $$\psi_2(x,y) = (\exp{2\pi ix},-y), x \in (\frac{3}{4},1].$$
Is this correct?
Thanks!