In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural generalization of finite groups with the discrete topology and have properties that carry over in significant fashion. Compact groups have a well-understood theory, in relation to group actions and representation theory. In the following we will assume all groups are Hausdorff spaces. Compact Lie groups Lie groups form a class of topological groups, and the compact Lie groups have a particularly well-developed theory. Basic examples of compact Lie groups include the circle group T and the torus groups Tn, the orthogonal group O(n), the special orthogonal group SO(n) and its covering spin group Spin(n), the unitary group U(n) and the special unitary group SU(n), the compact forms of the exceptional Lie groups: G2, F4, E6, E7, and E8. The classification theorem of compact Lie groups states that up to finite extensions and finite covers this exhausts the list of examples (which already includes some redundancies). This classification is described in more detail in the next subsection. Classification Given any compact Lie group G one can take its identity component G0, which is connected. The quotient group G/G0 is the group of components π0(G) which must be finite since G is compact. We therefore have a finite extension Meanwhile, for connected compact Lie groups, we have the following result: Theorem: Every connected compact Lie group is the quotient by a finite central subgroup of a product of a simply connected compact Lie group and a torus. Thus, the classification of connected compact Lie groups can in principle be reduced to knowledge of the simply connected compact Lie groups together with information about their centers. (For information about the center, see the section below on fundamental group and center.) Finally, every compact, connected, simply-connected Lie group K is a product of finitely many compact, connected, simply-connected simple Lie groups Ki each of which is isomorphic to exactly one of the following: The compact symplectic group The special unitary group The spin group or one of the five exceptional groups G2, F4, E6, E7, and E8. The restrictions on n are to avoid special isomorphisms among the various families for small values of n. For each of these groups, the center is known