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 $U V$ is an open set in $G$ .

4 - Let $G$ be a topological group and $K, L$ compact sets in $G$ . Then $K L$ 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	2013-03-22 17:37:38
Last modified on	2013-03-22 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