topgroups.tex

\subsection*{Extension: Topological Groups}

Prerequisite: to do this extension you should have taken either 
Topology or Real Analysis.

When groups appear in various areas of mathematics, they often have 
additional structure.  In Analysis and Geometry/Topology, this 
additional structure is often topological in nature, but this 
topological structure is consistent with the group operations.

\begin{definition}
  A \defn{topological group}{group!topological} is a group $(G, 
  \ast, e)$ together with a topology $\tau$ on $G$, such that 
  $\ast: G \cross G \to G$ and ${}^{-1} : G \to G$ are both 
  continuous.
\end{definition}

It turns out that it is sufficient to bundle the two conditions of 
this definition into one:

\begin{proposition}
  A group $(G, \ast, e)$ together with a topology $\tau$ on $G$, is a 
  topological group if and only if the function $(x,y) \mapsto 
  xy^{-1}$ is continuous from $G \cross G \to G$.
\end{proposition}


\begin{enumerate}
    \item Show that $(\reals, +, 0)$ is a topological group with its 
    usual topology.

    \item Show that $((0,\infty), \cdot, 1)$ is a topological group with its 
    usual topology.
    
    \item Find a homeomorphic group isomorphism between the 
    topological groups of the previous two problems.
    
    \item Prove Proposition.
\end{enumerate}