-
Notifications
You must be signed in to change notification settings - Fork 0
Commit
This commit does not belong to any branch on this repository, and may belong to a fork outside of the repository.
- Loading branch information
Showing
3 changed files
with
71 additions
and
214 deletions.
There are no files selected for viewing
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -1,192 +1 @@ | ||
|
|
||
| \chapter{Topological Group} | ||
| \section{Topological Group} | ||
| \begin{definition}{Topological Group}{} | ||
| A \textbf{topological group} is a group $G$ equipped with a topology $\tau$ such that the group multiplication map | ||
| \begin{align*} | ||
| \mu:G\times G&\longrightarrow G\\ | ||
| (g,h)&\longmapsto gh | ||
| \end{align*} | ||
| and the inversion map | ||
| \begin{align*} | ||
| \sigma:G&\longrightarrow G\\ | ||
| g&\longmapsto g^{-1} | ||
| \end{align*} | ||
| are continuous maps. | ||
| \end{definition} | ||
|
|
||
| \noindent Topological groups are the group objects in the category $\mathsf{Top}$. | ||
| \begin{definition}{Topological Group Category}{} | ||
| Topological groups form a category $\mathsf{TopGrp}$, where the morphisms are continuous group homomorphisms. | ||
| \end{definition} | ||
|
|
||
| \noindent An isomorphism of topological groups is a group isomorphism that is also a homeomorphism of the underlying topological spaces. | ||
| \begin{proposition}{}{} | ||
| The category $\mathsf{TopGrp}$ is complete. Limits in $\mathsf{TopGrp}$ commute with | ||
| \begin{itemize} | ||
| \item forgetful functor $\mathsf{TopGrp}\to\mathsf{Top}$ | ||
| \item forgetful functor $\mathsf{TopGrp}\to\mathsf{Grp}$ | ||
| \end{itemize} | ||
| \end{proposition} | ||
|
|
||
| \begin{prf} | ||
| It is enough to prove the existence and commutation for products and equalizers. Let $R_i, i \in I$ be a collection of topological rings. Take the usual product $R=\prod R_i$ with the product topology. Since $R \times R=\prod\left(R_i \times R_i\right)$ as a topological space (because products commutes with products in any category), we see that addition and multiplication on $R$ are continuous. Let $a, b: R \rightarrow R^{\prime}$ be two homomorphisms of topological rings. Then as the equalizer we can simply take the equalizer of $a$ and $b$ as maps of topological spaces, which is the same thing as the equalizer as maps of rings endowed with the induced topology. | ||
| \end{prf} | ||
|
|
||
|
|
||
| \begin{proposition}{Subgroups of Topological Group are Topological Groups}{} | ||
| Let $G$ be a topological group and $H$ be a subgroup of $G$. Then $H$ is a topological group with the subspace topology induced by $G$. | ||
| \end{proposition} | ||
|
|
||
| \begin{prf} | ||
| Since $H$ is a subgroup of $G$, the group multiplication map $\mu:G\times G\to G$ restricts to a map $\mu|_{H\times H}:H\times H\to H$. Since the inclusion $i:H\hookrightarrow G$ is continuous, | ||
| \[ | ||
| \mu|_{H\times H}:H \times H \xrightarrow{i\times i} G\times G\xrightarrow{\mu}\mu(G) \xrightarrow{i'} H | ||
| \] | ||
| is also continuous. Similarly, the inversion map $\sigma:G\to G$ restricts to a map $\sigma|_H:H\to H$ and $\sigma|_H$ is continuous. Hence $H$ is a topological group. | ||
|
|
||
| \end{prf} | ||
|
|
||
|
|
||
| \begin{proposition}{Translation Invariance}{} | ||
| For any $a \in G$, left or right multiplication by $a$ yields a homeomorphism $G \rightarrow G$. | ||
| \end{proposition} | ||
|
|
||
| \begin{prf} | ||
| Given any $a\in G$, let | ||
| \begin{align*} | ||
| L_a=\mu(a,\cdot):G&\longrightarrow G\\ | ||
| g&\longmapsto ag | ||
| \end{align*} | ||
| be the left multiplication map and | ||
| \begin{align*} | ||
| R_a=\mu(\cdot,a):G&\longrightarrow G\\ | ||
| g&\longmapsto ga | ||
| \end{align*} | ||
| be the right multiplication map. Since the group multiplication map $\mu:G\times G\to G$ is continuous, $L_a$ and $R_a$ must be continuous. Note that $L_a^{-1}=L_{a^{-1}}$ and $R_a^{-1}=R_{a^{-1}}$. Then we see $L_a^{-1}$ and $R_a^{-1}$ are also continuous maps. Hence $L_a$ and $R_a$ are homeomorphisms. | ||
| \end{prf} | ||
|
|
||
|
|
||
| \begin{corollary}{}{translation_invariance_cor} | ||
| Given any $a \in G$ and $S \subseteq G$, let's denote $a S:=\{a s: s \in S\}$ and $S a:=\{s a: s \in S\}$. Then | ||
| \begin{itemize} | ||
| \item $S$ is open $\iff$ $a S$ is open $\iff$ $a S$ is open. | ||
| \item $S$ is closed $\iff$ $a S$ is closed $\iff$ $a S$ is closed. | ||
| \end{itemize} | ||
| \end{corollary} | ||
|
|
||
| \begin{proposition}{Neighborhood Basis at $1_G$ Determines the Topology of $G$}{} | ||
| Given a topological group $G$, if $\mathcal{N}$ is a neighborhood basis of the identity element $1_G$, then for all $x \in X$, | ||
| \[ | ||
| x \mathcal{N}:=\{x N: N \in \mathcal{N}\} | ||
| \] | ||
| is a neighborhood basis of $x$ in $G$. In particular, the topology on $G$ is completely determined by any neighborhood basis at the identity element. | ||
| \end{proposition} | ||
|
|
||
| \begin{prf} | ||
| Let $x \in G$ and $V$ be any neighborhood of $x$. There exists an open set $U$ such that $x\in U\subseteq V$. By \Cref{th:translation_invariance_cor}, $x^{-1} U$ is an open neighborhood of $1_G$. Since $\mathcal{N}$ is a neighborhood basis of $1_G$, there exists $N \in \mathcal{N}$ such that $N \subseteq x^{-1} U$. Then there exists $x N \in x \mathcal{N}$ such that $x N \subseteq U$. Hence $x \mathcal{N}$ is a neighborhood basis of $x$. | ||
| \end{prf} | ||
|
|
||
|
|
||
|
|
||
| \begin{definition}{Inverse Limit in $\mathsf{TopGrp}$}{} | ||
| Let $\mathsf{I}$ be a \hyperref[th:filtered_category]{filtered} \hyperref[th:thin_category]{thin category} and $F:\mathsf{I}^{\mathrm{op}}\to \mathsf{TopGrp}$ be a functor. Similar to the \hyperref[th:inverse_limit_of_groups]{inverse limit in \textsf{Grp}}, we can unpack the information of $F$ into an inverse system $\left(\left(G_i\right)_{i \in I},\left(f_{i j}\right)_{i \leq j \in I}\right)$. The inverse limit of this inverse system is $\varprojlim F$, also denoted by $\varprojlim_{i\in I}G_i$.\\ | ||
| To give a concrete construction of $\varprojlim_{i\in I}G_i$, we can take the inverse limit of the underlying group and endow it with the subspace topology induced by the product topology on $\prod_{i\in I}G_i$. | ||
| \end{definition} | ||
|
|
||
|
|
||
| \section{Continuous Topological Group Action} | ||
| \begin{definition}{Compact Open Topology}{} | ||
| Let $X$ be a topological space and $K$ be a compact subset of $X$. The \textbf{compact open topology} on $\mathrm{Hom}_{\mathsf{Top}}(X,Y)$ is the topology generated by the subbasis | ||
| \[ | ||
| \mathcal{S}:=\left\{f\in \mathrm{Hom}_{\mathsf{Top}}(X,Y)\midv K\text{ is compact in }X,\;V\text{ is open in }Y,\;f(K)\subseteq V\right\}. | ||
| \] | ||
| \end{definition} | ||
|
|
||
|
|
||
| \begin{definition}{Group Action on Topological Space by Homeomorphisms}{} | ||
| A \textbf{group action} on a topological space $X$ is a group homomorphism $\rho:G\to \mathrm{Aut}_{\mathsf{Top}}(X)$, where $\mathrm{Aut}_{\mathsf{Top}}(X)$ is the group of all homeomorphisms from $X$ to itself. | ||
| \end{definition} | ||
|
|
||
|
|
||
| \begin{definition}{Continuous Topological Group Action on Topological Space}{} | ||
| A \textbf{continuous topological group action} on a topological space $X$ is a group homomorphism $\rho:G\to \mathrm{Aut}_{\mathsf{Set}}(X)$, where $G$ is a topological group, such that the following map induced by $\rho$ | ||
| \begin{align*} | ||
| \varrho:G\times X&\longrightarrow X\\ | ||
| (g,x)&\longmapsto \rho(g)(x) | ||
| \end{align*} | ||
| is continuous. In this case, we have $\mathrm{im}\rho \subseteq \mathrm{Aut}_{\mathsf{Top}}(X)$. | ||
| \end{definition} | ||
|
|
||
| \begin{prf} | ||
| For any $g\in G$, | ||
| \begin{align*} | ||
| \rho(g): X &\longrightarrow X\\ | ||
| x &\longmapsto \varrho(g,x) | ||
| \end{align*} | ||
| is continuous and has a continuous inverse $\rho(g^{-1})$. Hence $\rho(g)\in \mathrm{Aut}_{\mathsf{Top}}(X)$. | ||
| \end{prf} | ||
|
|
||
| From the definition, we see if $varpho:G\times X\to X$ is continuous topological group action on topological space $X$, it is also a group action on $X$ by homeomorphisms. If $G$ is discrete, then the converse holds. | ||
|
|
||
| \begin{proposition}{Discrete Group Acts Continuously on Topological Space $\iff$ Acts by Homeomorphisms}{} | ||
| Let $G$ be a group acting on the underlying set of a topological space $X$ through a group homomorphism $\rho:G\to \mathrm{Aut}_{\mathsf{Set}}(X)$. Then the following are equivalent: | ||
| \begin{enumerate}[(i)] | ||
| \item $G$ equipped with discrete topology acts continuously on $X$ | ||
| \item $G$ acts by homeomorphisms on $X$, i.e., | ||
| $\mathrm{im}\rho \subseteq \mathrm{Aut}_{\mathsf{Top}}(X)$ | ||
| \end{enumerate} | ||
| \end{proposition} | ||
| \begin{prf} | ||
| We only need to prove (ii)$\implies$ (i). For any open set $U\subseteq X$, we have | ||
| \begin{align*} | ||
| \varrho^{-1}(U)&=\{(g,x)\in G\times X\mid \varrho(g,x)\in U\}\\ | ||
| &=\{(g,x)\in G\times X\mid \rho(g)(x)\in U\}\\ | ||
| &=\{(g,x)\in G\times X\mid x\in \rho(g)^{-1}(U)\}\\ | ||
| &=\bigcup_{g\in G}\left(\left\{g\right\}\times \rho(g)^{-1}(U) \right) | ||
| \end{align*} | ||
| Since $\rho(g)$ is a homeomorphism, $\rho(g)^{-1}(U)$ is open for any open set $U$. Since $G$ is discrete, each $\left\{g\right\}\times \rho(g)^{-1}(U)$ is open in $G\times X$. Hence $\varrho^{-1}(U)$ as a union of open sets is open in $G\times X$, which implies $\varrho$ is continuous. | ||
| \end{prf} | ||
|
|
||
|
|
||
| \begin{definition}{Orbit Space}{} | ||
| Let $G$ be a group acting on a topological space $X$. The \textbf{orbit space} of $X$ under the action of $G$ is the quotient space $G\backslash X $ obtained by identifying all points in $X$ that are in the same orbit. $G\backslash X $ is equipped with the quotient topology: a subset $U\subseteq G\backslash X $ is open if and only if $\pi^{-1}(U)$ is open in $X$, where $\pi:X\to G\backslash X$ is the quotient map. | ||
| \end{definition} | ||
|
|
||
|
|
||
|
|
||
| \begin{proposition}{}{} | ||
| For any continuous action of a topological group $G$ on a topological space $E$, the quotient map $p: E \rightarrow G\backslash E$ is an open map. | ||
| \end{proposition} | ||
|
|
||
| \begin{prf} | ||
| For any $g \in G$ and any subset $U \subseteq M$, we define a set $g \cdot U \subseteq M$ by | ||
| $$ | ||
| g \cdot U=\{g \cdot x: x \in U\} . | ||
| $$ | ||
| If $U \subseteq M$ is open, then $\pi^{-1}(\pi(U))$ is equal to the union of all sets of the form $g \cdot U$ as $g$ ranges over $G$. Since $p \mapsto g \cdot p$ is a homeomorphism, each such set is open, and therefore $\pi^{-1}(\pi(U))$ is open in $M$. Becaues $\pi$ is a quotient map, this implies that $\pi(U)$ is open in $G\backslash M$, and therefore $\pi$ is an open map. | ||
| \end{prf} | ||
|
|
||
|
|
||
|
|
||
|
|
||
| \section{Topological Ring} | ||
| \begin{definition}{Topological Ring}{} | ||
| A \textbf{topological ring} is a ring $R$ equipped with a topology $\tau$ such that the ring addition map | ||
| \begin{align*} | ||
| +:R\times R&\longrightarrow R\\ | ||
| (a,b)&\longmapsto a+b | ||
| \end{align*} | ||
| the ring multiplication map | ||
| \begin{align*} | ||
| \cdot:R\times R&\longrightarrow R\\ | ||
| (a,b)&\longmapsto a\cdot b | ||
| \end{align*} | ||
| and the addition inverse map | ||
| \begin{align*} | ||
| -:R&\longrightarrow R\\ | ||
| a&\longmapsto -a | ||
| \end{align*} | ||
| are continuous maps. | ||
| \end{definition} | ||
|
|