A topological group is compactly generated in the algebraic sense when a compact subset generates its underlying group. Adjoining its inverse and the identity gives a compact symmetric generating set , and is a sigma-compact space. Some conventions restrict the term to locally compact groups or require a compact generating neighbourhood; those stronger conventions supply local compactness. Compact generation by an arbitrary compact subset alone does not.
The compact set generates the additive group , since every positive integer denominator divides some factorial. With its usual topology, is metrizable and is not locally compact. Nor can it have a nonzero Haar measure finite on compact sets: invariance gives equal masses to all singletons; a positive singleton mass makes have infinite mass, whereas zero singleton masses make the countable group have zero total mass.
// Target: analysis.bigb
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A **compactly generated group** is a type of topological group that can be characterized by the manner in which it is generated by compact subsets. Specifically, a topological group \( G \) is said to be compactly generated if there exists a compact subset \( K \subseteq G \) such that the whole group \( G \) can be expressed as the closure of the subgroup generated by \( K \).