A simplification of the Eilenberg-Steenrod axioms, 2 Original Research Article

Since the famous book of Eilenberg and Steenrod (1952) it is well known that a homology theory on the category of pairs of finite triangulable spaces and continuous maps is determined by four simple axioms. In 1988 Dawson showed that in the much smaller category of pairs of finite simplicial complexes and simplicial maps one can replace this system of axioms by a simpler one, i.e., by replacing the dimension axiom and omitting the homotopy axiom. In this paper I will show that it is also possible to do the same on the bigger category which was dealt with by Eilenberg and Steenrod. My proof uses ideas of Dawson, but it is easier. The second theorem shows that the homotopy axiom is nearly redundant.