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 $G$:

• $Ar:=\{ar:a\in A\}$

• $rA:=\{ra:a\in A\}$

• $AB:=\{ab:a\in A,b\in B\}$

• $A^2:=\{a_1{a}_2:a_1,a_2\in A\}$

• $A^{-1}:=\{a^{-1}:a\in A\}$

• $\overline{A}$ denotes the closure of $A$

$$

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 $ℬ$ be a neighborhood base around $e$. Then ${\{Br\}}_{B\in ℬ}$ is a neighborhood base around $r\in G$ and $\{Br:B\in ℬ\text{ and }r\in G\}$ 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.