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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.
MSC:
 55N40 Axioms for homology theory and uniqueness theorems in algebraic topology
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References:
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