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The Structure of σ-Ideals of Compact Sets

Kechris, A. S. and Louveau, A. and Woodin, W. H. (1987) The Structure of σ-Ideals of Compact Sets. Transactions of the American Mathematical Society, 301 (1). pp. 263-288. ISSN 0002-9947. doi:10.2307/2000338.

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Motivated by problems in certain areas of analysis, like measure theory and harmonic analysis, where σ-ideals of compact sets are encountered very often as notions of small or exceptional sets, we undertake in this paper a descriptive set theoretic study of σ-ideals of compact sets in compact metrizable spaces. In the first part we study the complexity of such ideals, showing that the structural condition of being a σ-ideal imposes severe definability restrictions. A typical instance is the dichotomy theorem, which states that σ-ideals which are analytic or coanalytic must be actually either complete coanalytic or else G_δ. In the second part we discuss (generators or as we call them here) bases for σ-ideals and in particular the problem of existence of Borel bases for coanalytic non-Borel σ-ideals. We derive here a criterion for the nonexistence of such bases which has several applications. Finally in the third part we develop the connections of the definability properties of σ-ideals with other structural properties, like the countable chain condition, etc.

Item Type:Article
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Additional Information:© 1987 American Mathematical Society. Received by the editors October 15, 1985. Partially supported by NSF Grant DMS-8416349.
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Issue or Number:1
Classification Code:1980 Mathematics Subject Classification (1985 Revision). Primary 03E15, 28A05, 28A12, 42A63
Record Number:CaltechAUTHORS:20130522-103643006
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Official Citation:The Structure of σ-Ideals of Compact Sets(pp. 263-288) A. S. Kechris, A. Louveau and W. H. Woodin Stable URL:
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:38631
Deposited By: Ruth Sustaita
Deposited On:22 May 2013 18:12
Last Modified:09 Nov 2021 23:38

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