finitely generated

Etymology
From the study of generators. The motivation for calling topologies satisfying "finitely generated" is that any topology satisfies  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

 * 1)  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:
 * 2)  Such that the functor $$\operatorname{Hom}_\mathcal{C}(X, \cdot)$$ preserves those filtered colimits of monomorphisms.
 * 3)  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.
 * 4)  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.)
 * 5)  Finitely generated as a (left) module over $$R$$.
 * 6)   Equipped with an  (i.e. one where the intersection of every family of open sets is open).