Compact Lie group

Remarks

Example: the special unitary group $\mathrm{SU}(2)$ is the set of complex matrices $\begin{pmatrix} a & -\bar b \\ b & \bar a \end{pmatrix}$ with $\abs{a}^2 + \abs{b}^2 = 1$. It is a closed subgroup of $\mathrm{GL}_2(\C)$, so it is a Lie group. As a space it is the unit sphere $S^3$ in $\C^2 \cong \R^4$, which is closed and bounded, hence compact.

Non-example: the additive group $\R$ is a Lie group, but it is not compact, since the open cover by the intervals $(-n, n)$ has no finite subcover.

A compact group need not be a Lie group. The group $\Z_p$ of $p$-adic integers is compact, but each neighbourhood of $0$ contains the nontrivial subgroup $p^k \Z_p$ for $k$ large, so it is not a Lie group.

Each compact Lie group is isomorphic to a closed subgroup of some unitary group $\mathrm{U}(n)$.