CH, a problem of Rolewicz and bidiscrete systems

Mirna Džamonja, István Juhász

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Abstract

We give a construction under CH of a non-metrizable compact Hausdorff space K such that any uncountable ‘nice’ semi-biorthogonal sequence in C(K) must be of a very specific kind. The space K has many nice properties, such as being hereditarily separable, hereditarily Lindelöf and a 2-to-1 continuous preimage of a metric space, and all Radon measures on K are separable. However K is not a Rosenthal compactum. We introduce the notion of a bidiscrete system in a compact space K. These are subsets of K2 which determine biorthogonal systems of a special kind in C(K) that we call nice. We note that for every infinite compact Hausdorff space K, the space C(K) has a bidiscrete system and hence a nice biorthogonal system of size d(K), the density of K.
Original languageEnglish
Pages (from-to)2485-2494
Number of pages10
JournalTopology and its Applications
Volume158
Issue number1
DOIs
Publication statusPublished - 1 Dec 2011

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