Generalized Steenrod homology theories are identical with partially continuous homology theories (Q721483)

Dropping the dimension axiom in Milnor's axiomatic characterization of the classic Steenrod homology leads to the concept of generalized homology theories which was introduced in [\textit{J. Kaminker} and \textit{C. Schochet}, Trans. Am. Math. Soc. 227, 63--107 (1977; Zbl 0368.46054)] and [\textit{D. S. Kahn} et al., Mich. Math. J. 24, 203--224 (1977; Zbl 0384.55001)]. In this work the author considers generalized homology theories restricted to various categories of compact metric spaces. Especially, he studies homology theories on the category of pairs of compacta \((CM^2)\) as a system \((h_n,\partial_n)\) of covariant homotopy invariant functors \(h_n:CM^2\to Ab\) (the category of abelian groups) and of natural transformations \(\partial_n:h_{n+1}\to h_n\) such that the exactness axiom and the excision axiom are satisfied and, additionally, the strong excision axiom and the cluster axiom, as well. Such a homology theory is called a Steenrod homology theory. The main result of this paper is that a Steenrod homology theory is equivalent to a maximally partially continuous homology theory, i.e. a homology theory satisfying a certain modification of the axiom of partial continuity which, originally, was introduced in [\textit{Kh. N. Inasaridze} and \textit{L. D. Mdzinarishvili}, Soobshch. Akad. Nauk Gruz. SSR 99, 317--320 (1980; Zbl 0448.55002)]. A homology theory on \(CM(CM^2)\) is said to be maximally partially continuous provided there exist natural transformations \(\beta_n:\varinjlim^1\circ h_{n+1}\to h_n\circ\varinjlim\) such that the corresponding short sequence \[ 0\to\varinjlim^1h_{n+1}(\mathbf X)\overset{\beta_n}{\rightarrow}h_n(X)\overset{p_\#}{\rightarrow}\varinjlim h_n(\mathbf {X)} 0 \] is exact for every \(n\in\mathbb N\) and every inverse sequence \(\mathbf X\) of (pairs) compacta.