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On the axiomatic systems of singular cohomology theory. (English) Zbl 1436.55004
Summary: On the category of pairs of topological spaces having the homotopy type of a CW complex the singular (co)homology theory was axiomatically studied by J. W. Milnor [Pac. J. Math. 12, 337–341 (1962; Zbl 0114.39604)]. In particular, Milnor gave an additivity axiom for a (co)homology theory and proved that any additive (co)homology theory on the given category is isomophic to the singular (co)homology. On the other hand, the singular homology is a homology with compact support [S. Eilenberg and N. Steenrod, Foundations of algebraic topology. Princeton: University Press, (1952; Zbl 0047.41402)]. In the paper [Trans. A. Razmadze Math. Inst. 172, No. 2, 265–275 (2018; Zbl 1397.55003)], L. Mdzinarishvili proposed partially compact support property for a cohomology theory and gave another axiomatic characterization of the singular cohomology theory. In this paper, we will give additional different axiomatic characterizations of the singular cohomology theory. Moreover, we will study connections of the mentioned axiomatic systems (cf. [A. Beridze and L. Mdzinarishvili, Topology Appl. 249, 73–82 (2018; Zbl 1447.55003)]).
MSC:
55N07 Steenrod-Sitnikov homologies
55N40 Axioms for homology theory and uniqueness theorems in algebraic topology
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