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# 数学代写|拓扑学代写TOPOLOGY代考|Quotients by Groups of Homeomorphisms

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## 数学代写|拓扑学代写TOPOLOGY代考|Quotients by Groups of Homeomorphisms

We indicate with $\operatorname{Homeo}(X)$ the set of homeomorphisms of a topological space $X$ to itself. The composition of maps induces on $\operatorname{Homeo}(X)$ a group structure, where the identity map is the neutral element.
Let $G \subset \operatorname{Homeo}(X)$ be a subgroup. Then
$x \sim y \quad$ if there exists $g \in G$ such that $y=g(x)$,for $x, y \in X$, defines an equivalence relation on $X$, whose cosets are called G-orbits, and whose quotient space is written $X / G$.

Proposition 5.15 Let $G \subset \operatorname{Homeo}(X)$ be a group of homeomorphisms of a space $X$, and $\pi: X \rightarrow X / G$ the canonical quotient map. Then $\pi$ is an open map. If $G$ is a finite group, $\pi$ is closed as well.
Proof For any subset $A \subset X$
$$\pi^{-1}(\pi(A))=\cup{g(A) \mid g \in G}$$
If $A$ is open, $g(A)$ is open for any $g \in G$, so $\pi^{-1}(\pi(A))$ is a union of open sets, hence open; by definition of quotient topology $\pi(A)$ is open.
In case $G$ is finite the same argument with $A$ closed remains valid.
Although even the quotient of a Hausdorff space by some group of homeomorphisms may not be Hausdorff (cf. Exercise 5.13), we have a better arsenal to decide whether a given quotient $X / \sim$ is Hausdorff in case we mod out by $G \subset \operatorname{Homeo}(X)$.

## 数学代写|拓扑学代写TOPOLOGY代考|Projective Spaces

Amongst the foremost examples of quotient spaces by groups of homeomorphisms we must include the real projective spaces $\mathbb{P}^n(\mathbb{R})$. By definition $\mathbb{P}^n(\mathbb{R})$ is the quotient of $\mathbb{R}^{n+1}-{0}$ by the group of homotheties (dilations): in other words the quotient under the equivalence relation
$$x \sim y \Longleftrightarrow x=\lambda y \text { for some } \lambda \in \mathbb{R}-{0}$$
Note that there’s a natural $1-1$ correspondence between $\mathbb{P}^n(\mathbb{R})$ and the set of 1-dimensional vector subspaces (lines) of $\mathbb{R}^{n+1}$ through the origin. Given a vector $\left(x_0, \ldots, x_n\right) \in \mathbb{R}^{n+1}-{0}$, the equivalence class in $\mathbb{P}^n(\mathbb{R})$ is usually written $\left[x_0, \ldots, x_n\right]$.

We put on $\mathbb{P}^n(\mathbb{R})$ the quotient topology; by Proposition 5.15 the canonical map $\pi: \mathbb{R}^{n+1}-{0} \rightarrow \mathbb{P}^n(\mathbb{R})$ is an open identification.

Composing the inclusion $i: S^n \rightarrow \mathbb{R}^{n+1}-{0}$ and $\pi$ produces a continuous, onto map that factorises through a continuous bijection $f: S^n / \sim \rightarrow \mathbb{P}^n(\mathbb{R})$, where for $x, y \in S^n$ we set $x \sim y$ iff $x= \pm y$. We claim that $f$ is a homeomorphism; to this end let’s consider the commutative diagram
\begin{aligned} & S^n / \sim \stackrel{f}{\longrightarrow} \mathbb{P}^n(\mathbb{R}) \quad \stackrel{f^{-1}}{\longrightarrow} S^n / \sim \ & \end{aligned}
Both the inclusion $i$ and the map $r$ are continuous, so also $\pi i$ and $\pi^{\prime} r$ are continuous. By Proposition 5.8, then, $f$ and $f^{-1}$ are continuous.

## 数学代写|拓扑学代写TOPOLOGY代考|Quotients by Groups of Homeomorphisms

$x \sim y \quad$，如果存在$g \in G$，使得$y=g(x)$对于$x, y \in X$在$X$上定义等价关系，其余集称为g轨道，其商空间为$X / G$。

$$\pi^{-1}(\pi(A))=\cup{g(A) \mid g \in G}$$

## 数学代写|拓扑学代写TOPOLOGY代考|Projective Spaces

$$x \sim y \Longleftrightarrow x=\lambda y \text { for some } \lambda \in \mathbb{R}-{0}$$

\begin{aligned} & S^n / \sim \stackrel{f}{\longrightarrow} \mathbb{P}^n(\mathbb{R}) \quad \stackrel{f^{-1}}{\longrightarrow} S^n / \sim \ & \end{aligned}

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