Let $E=\{([x],v)\in \Bbb RP^1\times \Bbb R^2: v\in [x]\}$ be the total space of the canonical line bundle over $\Bbb RP^1$. (Here $[x]$ denotes the line passing through the origin and $x\in \Bbb R^2-\{0\}$.) Consider its subspace $M=\{([x],v)\in E:|v|\leq 1\}$. Why is $M$ a Mobius band?
I am reading the book Characteristic Classes, and in p.94 of chapter 8, it is written that $M$ is obviously a Mobius band bounded by a circle. But I can't see why it is obvious.. (even in intuition)
Since $\mathbb{RP}^1 \cong S^1$ is a sphere line bundles are classified by their clutching functions, in particular isomorphism classes of line bundles are in bijective correspondence with $\pi_0(O(1))\cong \mathbb{Z}/2$. The isomorphism classes are distinguished by orientability, in particular $V \to S^1$ is trivial iff it is orientable.
We can explicitly describe a non-trivial bundle by writing down a clutching function. Let $U = S^1 - \{(0,1)\}$ and $V = S^1 - \{(0,-1)\}$ so that their intersection is two disjoint contractible components $U\cap V = C_+ \sqcup C_-$ where $C_+$ contains $(1,0)$ and $C_-$ contains $(-1, 0)$. Then a bundle's isomorphism class is determined by specifying a pointed function $\varphi\colon C_+ \sqcup C_-\to O(1)\cong \mathbb{Z}/2$, but since $\varphi$ is pointed its restriction to $C_+$ has constant value $1$, so the isomorphism class is determined by specifying any (not necessarily pointed) function $\varphi_-\colon C_- \to O(1)$. If we choose $\varphi_-$ to be constantly $-1$ then the bundle constructed via the clutching construction will be non-trivial by the classification theorem. But the bundle we get via the clutching construction is isomorphic to the one we get by taking the quotient of the space $[0,1] \times \mathbb{R}$ by the relation $(0,v) \sim (1, -v)$, and it is clear that this bundle has the Mobius band as its unit disk bundle.
But now the canonical bundle over $\mathbb{RP}^1$ is non-orientable and therefore isomorphic to the explicitly constructed bundles above, so its unit disk bundle is also the Mobius band.