Luzin space: Difference between revisions
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In [[real analysis]] and [[descriptive set theory]], a Luzin set (also Lusin set) is an [[uncountable]] set A such that every uncountable subset of A is of second [[Baire category]]. Equivalently, A is an uncountable set which meets every first category set in only countably many points. [[Nikolai Luzin|Luzin]] proved that such a set exists as a subset of every second category set. The proof requires the [[continuum hypothesis]]. |
In [[real analysis]] and [[descriptive set theory]], a Luzin set (also Lusin set) is an [[uncountable]] set A such that every uncountable subset of A is of second [[Baire category]]. Equivalently, A is an uncountable set which meets every first category set in only countably many points. [[Nikolai Luzin|Luzin]] proved that such a set exists as a subset of every second category set. The proof requires the [[continuum hypothesis]]. |
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Obvious properties of a Luzin set are that it must be |
Obvious properties of a Luzin set are that it must be of second category (otherwise the set itself is an uncountable first category subset) and of [[measure zero]], because every set of positive contains a first category set which also has positive measure, and is therefore uncountable. |
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The [[measure-category duality]] provides a measure analogue of Luzin sets - sets of positive measure every uncountable subset of which also has positive measure. |
The [[measure-category duality]] provides a measure analogue of Luzin sets - sets of positive measure every uncountable subset of which also has positive measure. |
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==References== |
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{{cite book |author=John C. Oxtoby |title=Measure and category: a survey of the analogies between topological and measure spaces |publisher=Springer-Verlag |location=Berlin |year=1980 |pages= |isbn=0-387-90508-1 |oclc= |doi=}} |
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Revision as of 21:26, 30 April 2007
In real analysis and descriptive set theory, a Luzin set (also Lusin set) is an uncountable set A such that every uncountable subset of A is of second Baire category. Equivalently, A is an uncountable set which meets every first category set in only countably many points. Luzin proved that such a set exists as a subset of every second category set. The proof requires the continuum hypothesis.
Obvious properties of a Luzin set are that it must be of second category (otherwise the set itself is an uncountable first category subset) and of measure zero, because every set of positive contains a first category set which also has positive measure, and is therefore uncountable.
The measure-category duality provides a measure analogue of Luzin sets - sets of positive measure every uncountable subset of which also has positive measure.
References
John C. Oxtoby (1980). Measure and category: a survey of the analogies between topological and measure spaces. Berlin: Springer-Verlag. ISBN 0-387-90508-1.
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