In mathematics, the term countably generated can have several meanings:

  • An algebraic structure (group, module, algebra) having countably many generators, see generating set
  • Countably generated space, a topological space in which the topology is determined by its countable subsets
  • Countably generated module. (Kaplansky's theorem says that a projective module is a direct sum of countably generated modules.)

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Countably generated space

convergent sequences. The countably generated spaces are precisely the spaces having countable tightness—therefore the name countably tight is used as well

Countably generated module

over a (not necessarily commutative) ring is countably generated if it is generated as a module by a countable subset. The importance of the notion comes

Kaplansky's theorem on projective modules

over an arbitrary ring is a direct sum of countably generated projective modules. Show that a countably generated projective module over a local ring is

Finitely generated module

...] of all polynomials in countably many variables. R itself is a finitely generated R-module (with {1} as generating set). Consider the submodule

Σ-algebra

Lebesgue measurable set is equivalent to some Borel set) but not countably generated (since its cardinality is higher than continuum). A separable measure

Finitely generated group

group is finitely generated, since S can be taken to be G itself. Every infinite finitely generated group must be countable but countable groups need not

Projective module

example is R/I where R is a direct product of countably many copies of F2 and I is the direct sum of countably many copies of F2 inside of R. The R-module

Standard probability space

\textstyle (\Omega ,{\mathcal {F}},P)} is standard if it is countably separated, countably generated, and absolutely measurable. See (Rokhlin 1952, the end