The "isomorphism" notion A typical mathematical field is usually described by a class of sets that are endowed with a "structure" concept that characterizes the subject matter of the field. Schematically, a typical branch of study in mathematics consists of pairs (X, 6), where X is a set and & is the "structure" imposed on the set X that is characteristic to the area. The structure & can be expressed in terms of algebraic or topological properties or a mixture of the two. Here are a few examples of mathematical areas. (1) Groups: Here for the typical object (X, S), the structure & repre- sents the algebraic structure on X given by the operation of mul- tiplication (x,y) i— xy and of the inverse function x \-+ x~l. (2) Topological spaces: Here for the typical object (X, ©), the structure & represents the topology of the set X. (3) Vector spaces: Here for the typical object (X, (3), the structure & represents the algebraic structure imposed on X by means of the addition (x, y) \-^ x + y and the scalar multiplication (A, x) i— Ax . (4) Topological vector spaces: Here for the typical object (X, 6) , the structure 6 represents the mixture of the vector space structure of X and the topological structure of X that makes the algebraic operations of X continuous. (5) Ordered vector spaces: Here for the typical object (X, 6) , the struc- ture & represents the algebraic structure of X together with the vector ordering on X. Once one deals with the objects of a mathematical field, one would like to have a way of identifying two objects of the field that look "alike." Xiii
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