In topology, one studies such objects as topological spaces, topological manifolds, differentiable manifolds, polyhedra, and so on. In the theory of transformation groups, one studies the symmetries of such objects, or generally subgroups of the full group of symmetries. Usually, the group of symmetries comes equipped with a naturally denned topology (such as the compact-open topology) and it is important to consider this topology as part of the structure studied. In some cases of importance, such as the group of isometries of a compact riemannian manifold, the group of symmetries is a compact Lie group. This should be sufficient reason for studying compact groups of transformations of a space or of a manifold. An even more compelling reason for singling out the case of compact groups is the fact that one can obtain many strong results and tools in this case that are not available for the case of noncompact groups. Indeed, the theory of compact transformation groups has a completely different flavor from that of noncompact transformation groups.
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