basic results in topological groups
The purpose of this entry is to list some and useful results concerning the topological of topological groups^{}. We will use the following notation whenever $A,B$ are subsets of a topological group $G$ and $r$ an element of $G$:

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$Ar:=\{ar:a\in A\}$

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$rA:=\{ra:a\in A\}$

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$AB:=\{ab:a\in A,b\in B\}$

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${A}^{2}:=\{{a}_{1}{a}_{2}:{a}_{1},{a}_{2}\in A\}$

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${A}^{1}:=\{{a}^{1}:a\in A\}$

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$\overline{A}$ denotes the closure^{} of $A$
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1  Let $G$ be a topological group and $r\in G$. The left multiplication $s\mapsto rs$, multiplication $s\mapsto sr$, and inversion $s\mapsto {s}^{1}$, are homeomorphisms^{} of $G$.
2  Let $G$ be a topological group and $e\in G$ the identity element^{}. Let $\mathcal{B}$ be a neighborhood base around $e$. Then ${\{Br\}}_{B\in \mathcal{B}}$ is a neighborhood base around $r\in G$ and $\{Br:B\in \mathcal{B}\text{and}r\in G\}$ is a basis (http://planetmath.org/BasisTopologicalSpace) for the topology^{} of $G$.
3  Let $G$ be a topological group. If $U\subseteq G$ is open and $V$ is any subset of $G$, then $UV$ is an open set in $G$.
4  Let $G$ be a topological group and $K,L$ compact sets in $G$. Then $KL$ is also compact.
5  Let $G$ be a topological group and $e\in G$ the identity element. If $V$ is a neighborhood^{} of $e$ then $V\subset \overline{V}\subset {V}^{2}$.
6  Let $G$ be a topological group, $e\in G$ the identity element and $W$ a neighborhood around $e$. Then there exists a neighborhood $U$ around $e$ such that ${U}^{2}\subset W$.
7  Let $G$ be a topological group, $e\in G$ the identity element and $W$ a neighborhood around $e$. Then there exists a symmetric (http://planetmath.org/SymmetricSet) neighborhood $U$ around $e$ such that ${U}^{2}\subseteq W$.
8  Let $G$ be a topological group. If $H$ is a subgroup^{} of $G$, then so is $\overline{H}$.
9 Let $G$ be a topological group. If $H$ is an open subgroup of $G$, then $H$ is also closed.
Title  basic results in topological groups 

Canonical name  BasicResultsInTopologicalGroups 
Date of creation  20130322 17:37:38 
Last modified on  20130322 17:37:38 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  16 
Author  asteroid (17536) 
Entry type  Result 
Classification  msc 22A05 
Related topic  PolishGSpace 
Related topic  PolishGroup 