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

Abstract

The Alexandroff-\v{C}ech 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 \(\mathcal{K}^2_{Pol}\) to the category of closed normally embedded, the so called, \(P\)-pairs of general topological spaces \(\mathcal{K}^2_{Top}\). 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-\v{C}ech 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 \(\mathcal{K}^2_{Top},\) which is isomorphic to the Steenrod homology theory on the subcategory of compact pairs \(\mathcal{K}^2_{C}.\) Moreover, we give an axiomatic characterization of the constructed homology theory.