\chapter{Topological groups}

As mentioned, groups model clases of composable bijective functions particularly well and are hence an important algebraic structure in describing how spaces may be transformed (by said bijective functions).

This proves incredibly useful for many uses, including the study of continuous symmetry (Lie theory goes further to study smooth symmetry), applications in functional analysis and harmonic analysis, and fundamental in representation theory.

\begin{definition}[Topological group]
Group that is also a topological space
\end{definition}