On the axioms of homology theory

We give an axiomatization for homology and cohomology theory in the categories $\mathscr A$ and $\mathscr B$ of countable locally finite polyhedra and of locally compact metrizable spaces, respectively, with proper mappings; in the category $\mathscr B_0$ of metrizable compacta and continuous mappings; and (for cohomology) in the category $\mathscr B$ of locally compact metrizable spaces and arbitrary continuous mappings. In $\mathscr B$ we determine the kernel of the natural homomorphism $\varphi\colon H^n(X)\to\varprojlim H^n(C)$ over compact $C$ for a $\Pi$-additive cohomology (in particular, for Aleksandrov–Čech cohomology). Finally, we analyze the axioms of Sklyarenko (Math. Sb. (N.S.) 85(127) (1971), 201–223).
Bibliography: 6 titles.