C-18 - Eberlein Compacta

Abstract

Eberlein compacta were introduced to mathematics as compact subspaces of Banach spaces with weak topology. They have a natural description in terms of spaces of functions with the point wise topology. Arguments involving compact sets of functions play a key role in functional analysis. Another important property of Eberlein compacta requires the closure of any countable subset in an Eberlein compactum to be metrizable. Every separable Eberlein compactum is metrizable. It follows that any Tychonoff or Cantor cube of uncountable weight can serve as an example of a non-Eberlein compactum. H. Rosenthal proved that every Eberlein compactum with the countable Souslin number is metrizable. This considerably improves the statement on metrizability of separable Eberlein compacta and implies that every non-metrizable compact topological group is a non-Eberlein compactum. Every metrizable compactum is Eberlein. It is the one point compactification of an uncountable discrete space. Every Hausdorff continuous image of an Eberlein compactum is again an Eberlein compactum. Monolithicity of Eberlein compacta has another nontrivial corollary. A compact Hausdorff space that is the union of three metrizable subspaces need not be Eberlein.