On Axiomatic Characterization of Alexander-Spanier Normal Homology Theory of General Topological Spaces

Abstract

The Alexandroff-Čech normal cohomology theory [Mor${}_1$], [Bar], [Ba${}_1$],[Ba${}_2$] is the unique continuous extension \cite{Wat} of the additive cohomology theory [Mil], [Ber-Mdz${}_1$] from the category of polyhedral pairs ${\mathcal{K}}_{Pol}^2$ to the category of closed normally embedded, the so called, $P$-pairs of general topological spaces ${\mathcal{K}}_{Top}^2$. In this paper we define the Alexander-Spanier normal cohomology theory based on all normal coverings and show that it is isomorphic to the Alexandroff-Čech normal cohomology. Using this fact and methods developed in [Ber-Mdz${}_3$] we construct an exact, the so called, Alexander-Spanier normal homology theory on the category ${\mathcal{K}}_{Top}^2,$ which is isomorphic to the Steenrod homology theory on the subcategory of compact pairs ${\mathcal{K}}_C^2.$ Moreover, we give an axiomatic characterization of the constructed homology theory.