On products of quasi-F-compacta

Abstract

A Hausdorff compact space is called a quasi-F-compactum if it admits a decomposition into a special well-ordered inverse system with almost fully closed neighboring projections. Each Fedorchuk compactum (or F-compactum) is a quasi-F-compactum. We prove that for any uncountable cardinal number λ there exists an F-compactum X of spectral height λ, all finite powers of which are F-compacta of the same spectral height. Thus, the analogue of the assertion on the anti-multiplicativity of the class of F-compacta of spectral height 3 is false for F-compacta of uncountable spectral height.The product of a quasi-F-compactum (F-compactum) onto a countable compactum is always a quasi-F-compactum (F-compactum). At the same time, the product of a quasi-F-compactum of the spectral height 3 to an uncountable metrizable compactum is never a quasi-F-compactum of a countable spectral height.We also prove a number of assertions about almost fully closed mappings of products of quasi-F-compacta to metrizable compacta.