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Čech Cohomology Theory and the Axioms

In: The Mathematical Legacy of Eduard Čech

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  • C. H. Dowker

Abstract

Čech cohomology theory based on inimite coverings by open sets has proved useful in studying the homotopy classes of maps of general spaces.1 The purpose of the present paper is to show that this Čech cohomology theory satisfies the Eilenberg-Steenrod axioms.2 (As is known, the Čech cohomology theory based on finite coverings fails in general2 to satisfy axiom 4, the homotopy axiom.) The axioms are stated in terms of the relative cohomology groups of a pair (X, A) where A is any subset of a topological space X. However the relative Öech groups are usually defined only relative to closed subsets. In part C we extend the definition of relative Öech cohomology groups to the case of non-closed subsets. Part E contains the proof that the groups so defined satisfy the axioms for arbitrary pairs (X, A).

Suggested Citation

  • C. H. Dowker, 1993. "Čech Cohomology Theory and the Axioms," Springer Books, in: Miroslav Katětov & Petr Simon (ed.), The Mathematical Legacy of Eduard Čech, pages 318-332, Springer.
    • Handle: RePEc:spr:sprchp:978-3-0348-7524-0_24
    DOI: 10.1007/978-3-0348-7524-0_24
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