Compact group

Remarks

Example: the unitary group $U(n) = \set{A \in M_n(\C) : A^* A = I}$ with the topology from $M_n(\C) \cong \C^{n^2}$ is a compact group. It is Hausdorff as a subspace of $\C^{n^2}$. It is closed, as the zero set of the continuous map $A \mapsto A^*A - I$. It is bounded, since each column of a unitary matrix has norm $1$, so it is compact by the Heine–Borel theorem. Non-example: the group $\R$ with addition and the usual topology is a Hausdorff topological group. It is not compact, since the open cover by the intervals $(-n, n)$ has no finite subcover. Every finite group with the discrete topology is a compact group. Some authors do not require the Hausdorff condition; others include it in the definition of a topological group.