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On the continuity property of the exact homology theories. (English) Zbl 1441.55005
Summary: It is well known [S. Eilenberg and N. Steenrod, Foundations of algebraic topology. Princeton: University Press (1952; Zbl 0047.41402)] that on the category \(\mathcal{K}_c\) of compact pairs and continuous maps there is a continuity property of the partially exact homology [op. cit., Definition 2.3.X]. Namely, if the compact pair \((X, A)\) is an inverse limit of the compact pairs \((X_\alpha, A_\alpha)\), then the partially exact homology \(H_*\) of \((X, A)\) is an inverse limit of homology groups \((X_\alpha, A_\alpha)\); i.e. there is an isomorphism \[ H_*(X,A)\stackrel{\sim}\to\varprojlim H_*(X_\alpha, A_\alpha). \] It has been shown that among all the partially exact theories on the category \(\mathcal{K}_C\), the Čech theory is essentially the only one satisfying this continuity axiom [op. cit., Theorem 3.1.X].
We define a continuity property of the exact homology theories on the category \(\mathcal{K}_C\) and prove that the homology theory on the category \(\mathcal{K}_C\), satisfying all the Eilenberg-Steenrod axioms and the continuity property of the exact homology theories, exists.
55N40 Axioms for homology theory and uniqueness theorems in algebraic topology
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