On fully closed mappings of Fedorchuk compacta

An F-compactum or a Fedorchuk compactum is a compact Hausdorff topological space that admits a decomposition into a special fully ordered inverse spectrum with fully closed neighboring projections. F-compacta of spectral height 3 are exactly nonmetrizable compacta that admit a fully closed mapping onto a metric compactum with metrizable fibers. In this paper, it is proved that such a fully closed mapping for an F-compactum X of spectral height 3 is defined almost uniquely. Namely, nontrivial fibers of any two fully closed mapping of X into metric compacts with metrizable inverse images of points coincide everywhere, with a possible exception of a countable family of elements. Examples of F-compacta of spectral height 3 are, for example, Aleksandrov’s "two arrows" and the lexicographic square of the segment. It follows from the main result of this paper that almost all non-trivial layers of any admissible fully closed mapping are colons that are glued together under the standard projection of D onto the segment. Similarly, almost all nontrivial fibers of any admissible fully closed mapping necessarily coincide with the "vertical segments" of the lexicographic square.

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