Stability of almost commutative inverse systems of compacta
In a recent publication [Pac. J. Math. 138, No.1, 129-144 (1989; Zbl 0631.54006)] the first author and \textit{L. R. Rubin} introduced the notion of approximate inverse system of (metric) compacta over arbitrary directed set of indices where is not required the exact commutativity for bonding maps but only a commutativity up to a system of epsilons such that some natural conditions are satisfied. The notion of the limit for such systems was introduced. The authors introduce inverse systems called almost commutative which are potentially approximate, which means that the epsilons are not explicitly given, but they can be chosen such that the approximate systems arise. The limit of an approximate system arising from a given almost commutative system occurs to be not dependent on the above mentioned choice of system of epsilons. So this limit is by definition the limit of the given almost commutative system. There is proved that ``sufficiently small perturbations of the bonding maps of the almost commutative system yield again an almost commutative system with the same limit space''. In the case when bonding maps are near homeomorphisms (i.e. if they are arbitrarily close to homeomorphisms), they can be changed after mentioned above perturbations to homeomorphisms. Although the projections from the limit space are not necessarily homeomorphisms, they are refinable maps (a non-metric analogue of near homeomorphisms). Note that the limit space is not necessarily metric and no assumption on the cardinalities of sets of indices are made.
scientific article; zbMATH DE number 4094347
scientific article