mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a compactly generated (topological) group is a

topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two str ...

''G'' which is algebraically generated by one of its

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

subsets.. This should not be confused with the unrelated notion (widely used in

algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

) of a

compactly generated space In topology, a compactly generated space is a topological space whose topology is coherent with the family of all compact subspaces. Specifically, a topological space ''X'' is compactly generated if it satisfies the following condition: :A subspa ...

-- one whose

topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

is generated (in a suitable sense) by its compact subspaces.

Definition

''G'' is said to be compactly generated if there exists a compact subset ''K'' of ''G'' such that :

\langle K\rangle = \bigcup_ (K \cup K^)^n = G.

So if ''K'' is symmetric, i.e. ''K'' = ''K''⁻¹, then :

G = \bigcup_ K^n.

Locally compact case

This property is interesting in the case of

locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...

topological groups, since locally compact compactly generated topological groups can be approximated by locally compact, separable

metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathema ...

factor groups of ''G''. More precisely, for a sequence :''U''_''n'' of open identity neighborhoods, there exists a

normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...

''N'' contained in the intersection of that sequence, such that :''G''/''N'' is locally compact metric separable (the Kakutani-Kodaira-Montgomery-Zippin theorem).

References

Topological groups {{topology-stub