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Reference.org
Compact group
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a compact (topological) group is a <a href="/facts/Topological_group/cAgNMMRU">topological group</a> whose <a href="/facts/Topology/2EqW47cU">topology</a> realizes it as a <a href="/facts/Compact_space/d0cgXJH7">compact topological space</a> (when an element of the group is operated on, the result is also within the group). Compact groups are a natural generalization of <a href="/facts/Finite_group/vv2tvogB">finite groups</a> with the <a href="/facts/Discrete_topology/hpvXA23u">discrete topology</a> and have properties that carry over in significant fashion. Compact groups have a well-understood theory, in relation to <a href="/facts/Group_action_(mathematics)/Vh9VtUUw">group actions</a> and <a href="/facts/Representation_theory/3ydLtaGH">representation theory</a>. </p><p>In the following we will assume all groups are <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff spaces</a>. </p>