Sets, The Direct Sum of Sets

The Direct Sum of Sets

Assume the components of an indexed set each have a designated member. This is usually zero, or the origin, or the identity element, or some "special" member of the set. The direct sum is the direct product with the following restriction. A function f is in the direct sum if f(x) is the "pointed" member in A_x, most of the time. For finitely many values of x, f(x) may differ from the designated point.

Consider the direct sum of infinitely many copies of the integers, where 0 is the designated element in each set. The sequence 0,0,3,4,0,7,9,0,0,0… is in the direct sum, assuming the zeros go on forever. However, the sequence 1,1,1,1…, with ones going on forever, is not. A finite number of entries can be nonzero; everything else must be zero.

We don't need the axiom of choice here, since a well defined function f maps every x to its point. In the previous example, f(x) = 0 for all x. The direct product, and the direct sum, is always nonempty.