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 \-+
(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."