A note on approximate systems of metric compacta

AbstractApproximate inverse systems and their limits were first defined for systems of metric compacta. Then the notion was extended to systems of arbitrary spaces. In both cases the usual commutativity requirement paa′pa′a″ = paa″, a⩽a′⩽a″, on the bonding mappings was replaced by a weaker condition, which allows the mappings paa′pa′a″ and paa″ to differ, though in a controlled way. In the first case the difference is measured by numbers εa > 0, while in the second case it is measured by normal coverings Ua of Xa. In the present paper one considers only systems of metric compacta and compares the two definitions in this case. Surprisingly, they are not equivalent and the first one can actually depend on the choice of the metrics