finitely generated

English

Etymology

From the study of generators. The motivation for calling topologies satisfying sense 1.5 "finitely generated" is that any topology satisfies sense 1.5 if and only if it is coherent with its finite subspaces. Thus, metaphorically, it is "generated" by them. The category-theoretic senses were created to generalize those of abstract algebra, and so were named identically.

Adjective

finitely generated (not comparable)

1. (algebra) In any of several specific senses, such that all its elements can be created using (or described by reference to) a finite set of elements, usually called generators:
   1. (most generally, category theory, of an object ${\displaystyle X}$ in a locally small category ${\displaystyle {\mathcal {C}}}$ that admits filtered colimits of monomorphism) Such that the functor ${\displaystyle \operatorname {Hom} _{\mathcal {C}}(X,\cdot )}$ preserves those filtered colimits of monomorphisms.
   2. (less generally, but still covering most cases outside pure category theory, of an object ${\displaystyle X}$ in a concrete category) Being a quotient object of a free object over a finite set, i.e. being the target of a regular epimorphism from an object which is free on a finite set.
   3. (abstract algebra, of a group-like structure: a group, module, ring, monoid, etc.) Having a finite set of generators, i.e. having a finite set of elements from which all other elements can be created in finitely many steps under the permitted operations (viz. the group operation for groups, addition and scalar multiplication for modules, addition and multiplication for rings, etc.)
   4. (ring theory, of a (left) ideal in a ring ${\displaystyle R}$) Finitely generated as a (left) module over ${\displaystyle R}$.
   6. (of a space) Equipped with an Alexandrov topology (i.e. one where the intersection of every family of open sets is open).