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

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