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

• $Ar := \{ar: a \in A\}$
• $rA := \{ra: a \in A\}$
• $AB:=\{ab: a \in A,\, b \in B\}$
• $A^2 := \{a_1a_2: a_1,a_2 \in A\}$
• $A^{-1} := \{a^{-1}: a\in A\}$
• $\overline{A}$ denotes the closure of $A$

Proposition 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 $\{Br:B\in \mathcal{B} { and }\, r \in G \}$ is a basis for the topology of $G$

Proposition 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 $G$

Proposition 4 - Let $G$ be a topological group and $K, L$ compact sets in $G$ Then $KL$ is also compact.

Proposition 5 - Let $G$ be a topological group and $e \in G$ the identity element. If $V$ is a neighborhood of $e$ then $V \subset \overline{V} \subset V^2$

Proposition 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 $U^2 \subset W$

Proposition 7 - Let $G$ be a topological group, $e \in G$ the identity element and $W$ a neighborhood around $e$ Then there exists a symmetric neighborhood $U$ around $e$ such that $U^2\subseteq W$

Proposition 8 - Let $G$ be a topological group. If $H$ is a subgroup of $G$ then so is $\overline{H}$

Proposition 9- Let $G$ be a topological group. If $H$ is an open subgroup of $G$ then $H$ is also closed.