We show that a compact space is I-favorable if, and only if it can be represented as the limit of a σ -complete inverse system of compact metrizable spaces with skeletal bonding maps. We also show that any completely regular I-favorable space can be embedded as a dense subset of the limit of a σ -complete inverse system of separable metrizable spaces with skeletal bonding maps.

We introduce the notion of a center of distances of a metric space and use it in a generalization of the theorem by John von Neumann on permutations of two sequences with the same set of cluster points in a compact metric space. This notion is also used to study sets of subsums of some sequences of positive reals, as well for some impossibility proofs. We compute the center of distances of the Cantorval, which is the set of subsums of the sequence 3 4 ,

We introduce and investigate the class of skeletally Dugundji spaces as a skeletal analogue of Dugundji space.Our main result states that the following conditions are equivalent for a given space X : (i) X is skeletally Dugundji; (ii) every compactification of X is co-absolute to a Dugundji space; (iii) every C * -embedding of the absolute (X ) in another space is strongly π-regular; (iv) X has a multiplicative lattice in the sense of Shchepin [Shchepin E.V., Topology of limit spaces with uncountable inverse spectra, Uspekhi Mat. Nauk, 1976, 31(5), 191-226 (in Russian)] consisting of skeletal maps. MSC:54C10, 54F65