$\mathbb{R}^2$的直线的空间 同胚于 开Möbius带

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21Sheet1

9. Every line in the real affine plane $\mathbb{R}^2$ can be written in the form $a x+b y+c=0$ where $a, b$ are not both zero. Of course, $\lambda a x+\lambda b y+\lambda c=0$ is an equation of the same line where $\lambda \neq 0$. Hence the space of lines can be identified with region.
$$
M=\frac{\mathbb{R}^2 \backslash\{(0,0)\} \times \mathbb{R}}{\mathbb{R}^*}
$$
Identify $M$ as a subspace of $\mathbb{RP}^2$. What is the topology or $M$?

因为$\lambda a x+\lambda b y+\lambda c=0$与$a x+b y+c=0$为同一条直线,
所以可以把直线$a x+b y+c=0$对应于$\mathbb{RP}^2={\mathbb R^3\over\mathbb R^*}$的点$[a,b,c]$.
但是$a,b$不同为0,需要去掉{(0,0)}
所以应该对应于$M=\frac{\mathbb{R}^2 \backslash\{(0,0)\} \times \mathbb{R}}{\mathbb{R}^*}$
还剩最后一问$M$的拓扑

相关: $\mathbb{RP}^2$同胚于Möbius带粘在圆盘上. MSE
以及
Exercise 22.4. Describe the homeomorphism $X → M$ described above explicitly.

compute the fundamental group of $\mathbb{R P}^2$

Recall the characterization of of $\mathbb{R P}^2$ from Figure 22.2, and denote by $e^2$ the interior of the disk. Consider a cover of $\mathbb{R P}^2$ as follows. Consider an open disk $B \subset e^2$ in $\mathbb{R} \mathbb{P}^2$ and a closed disk $C \subset B$, and define $A=\mathbb{R} \mathbb{P}^2-C$ (see Figure 22.3). Then $\mathbb{R P}^2=A \cup B$.

Fix a base point $x_0 \in A \cap B$. Clearly, $\pi_1\left(B, x_0\right)=\mathbf{1}$, the trivial group, since $B$ is just an open disk. The intersection $A \cap B$ is homotopic to a circle, represented by a loop $\omega$, so that $\pi_1\left(A \cap B, x_0\right)=\langle[\omega]\rangle \cong \mathbb{Z}$. The set $A$, in turn, is the interior of a Möbius strip, as seen in Example 22.3, with $\gamma$ representing the inner circle. As seen in Example $22.1, A$ deformation retracts to $\gamma$ (or, more precisely, to a circle homotopic to $\gamma$ but with basepoint $x_0$, see the figure), so that $\pi_1\left(A, x_0\right) \cong\langle[\gamma]\rangle \cong \mathbb{Z}$.

Since the fundamental group $\pi_1\left(B, x_0\right)$ is trivial, the free group $\pi_1\left(A, x_0\right) *$ $\pi_1\left(B, x_0\right)$ is generated by $[\gamma]$. To get the fundamental group of $\mathbb{R P}^2$ using Seifert-van Kampen, we have to factor out elements that are multiples of
$$
(\iota A \cap B)_*([\omega])
$$
where $\iota_{A \cap B}$ is the inclusion of $A \cap B$ in $A$. We can think of $\omega$ as the outer circle of a Möbius strip, and $\gamma$ as the inner circle. Going around $\omega$ once corresponds to going around $\gamma$ twice, so that
$$
\left(\iota_{A \cap B}\right)_*([\omega])=[\gamma]^2
$$
By the Seifert-van Kampen Theorem,
$$
\pi_1\left(\mathbb{R P}^2, x_0\right) \cong\langle[\gamma]\rangle /\left\langle[\gamma]^2\right\rangle \cong \mathbb{Z} / 2 \mathbb{Z}
$$

$$M=\mathbb{RP}^2 \setminus\{[0,0]\}$$
$\mathbb{RP}^2$去除一个点的空间$M$, 同胚于$\mathbb{RP}^2$去除一个圆盘的空间, 根据上面结论得$M$同胚于Möbius band
我有一个问题:
如何显式写出$M$中的开集是哪些？
也就是，Möbius带的开集映射到哪些直线的集合？

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Page 21
Finally, we can use duality to understand something very down-to-earth – the space of straight lines in $\mathbf{R}^2$. When we viewed the projective plane $P^2(\mathbf{R})$ as $\mathbf{R}^2 \cup P^1(\mathbf{R})$ we saw that a projective line not equal to the line at infinity $P^1(\mathbf{R})$ intersected $\mathbf{R}^2$ in an ordinary straight line. Since we now know that the lines in $P^2(\mathbf{R})$ are in one-to-one correspondence with another projective plane – the dual plane – we see that we only have to remove a single point from the dual plane, the point giving the line at infinity, to obtain the space of lines in $\mathbf{R}^2$. So in the sphere model, we remove the north and south poles and identify antipodal points.
Concretely parametrize the sphere in the usual way:
$$
x_1=\sin \theta \sin \phi, \quad x_2=\sin \theta \cos \phi, \quad x_3=\cos \theta
$$
then with the poles removed the range of values is $0<\theta<\pi, 0 \leq \phi<2 \pi$. The antipodal map is
$$
\theta \mapsto \pi-\theta, \quad \phi \mapsto \phi+\pi
$$
We can therefore identify the space of lines in $\mathbf{R}^2$ as the pairs
$$
(\theta, \phi) \in(0, \pi) \times[0, \pi]
$$
where we identify $(\theta, 0)$ with $(\pi-\theta, \pi)$ :

  and this is the Möbius band.