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 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

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