CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
A real Lie group is compact if its underlying topological group is a compact topological group.
Compact Lie groups have a very well understood structure theory.
All maximal tori of a compact Lie group are conjugate by inner automorphisms. The dimension of a maximal torus $T$ of a compact Lie group is called a rank of $G$. The normalizer $N(T)$ of a maximal torus $T$ determines $G$. The Weyl group $W(G)=W(G,T)$ of $G$ with respect to a choice of a maximal torus $T$ is the group of automorphisms of $T$ which are restrictions of inner automorphisms of $G$. The maximal torus is of finite index in its normalizer; the quotient $N(T)/T$ is isomorphic to $W(G)$. The cardinality of $W(G)$ for a compact connected $G$, equals the Euler characteristic of the homogeneous space $G/T$ (“flag variety”).
See also at relation between compact Lie groups and reductive algebraic groups