×

A short proof of a classical result of M. G. Tkachenko. (English) Zbl 1083.54017

Summary: We give a comparatively short proof of a theorem which states that any compact space that is a continuous image of a dense subspace of a \(\Sigma\)-product of spaces with countable network, is metrizable. This is a very deep and non-trivial result of M.G. Tkachenko obtained in 1982. The original proof consists of eight pages of a very compressed text which involves construction of an inverse system whose elements are also constructed using additonal inverse systems. We give a transparent proof on less than three pages hoping to contribute to a better understanding of the features of dense subspaces of \(\Sigma\)-products responsible for metrizability of their compact continuous images.

MSC:

54E35 Metric spaces, metrizability
54C05 Continuous maps
54D30 Compactness
PDFBibTeX XMLCite