R-MODULES FOR THE ALEXANDER COHOMOLOGY THEORY APPROVED: Major Professor

The Alexander Wallace Spanier cohomology theory associates with an arbitrary topological space an abelian group. In this paper, an arbitrary topological space is associated with an R-module. The construction of the R-module is similar to the Alexander Wallace Spanier construction of the abelian group. In Chapter I, the algebraic and topological material necessary to construct the theor'y is listed. This material is found in most graduate level algebx^a or topology courses. The definition of the cohomology R-modules is given in Chapter II. Also, it is verified that this construction does give a cohomology theory. The axioms necessary for this are proved as Theorems 1-7. In Chapter III, additional theorems extend the theory. Some applications of the theory are given in Chapter IV. One of the applications shows the exisrence of floors for each element of HP(X,A) if X is compact Hausdorff, and if A is a closed subset of X. The modules of the n-cell and the n-sphere are also given.