or quotient space, n. the result of using the structure of a given set (when that is possible) to impose a similar structure on the set of equivalence classes with respect to a given equivalence relation. For example, the factor group (or quotient group) G/H of a group G by a normal subgroup H is the set of cosets of H in G, and the factor ring (or quotient ring) R/K of a ring R by an ideal K is the set of cosets of K in R; one may similarly define factor topological groups, and, if the equivalence relation corresponds to membership in a vector subspace, one may construct factor vector spaces, factor normed or Banach spaces, and factor Hilbert spaces, where the subspace in these cases lies in the same class. See also three-space property.