Nowhere dense set: Difference between revisions
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An [[infinite set|infinite]] [[Cartesian product]] of non-empty non-[[compact space]]s has the property that every compact subset is nowhere dense. | An [[infinite set|infinite]] [[Cartesian product]] of non-empty non-[[compact space]]s has the property that every compact subset is nowhere dense. | ||
A finite [[union]] of nowhere dense sets is again nowhere dense. | |||
A '''first category space''' or '''meagre space''' is a [[countability|countable]] [[union]] of nowhere dense sets: any other topological space is of '''second category'''. The ''[[Baire category theorem]]'' states that a non-empty [[complete metric space]] is of second category. | A '''first category space''' or '''meagre space''' is a [[countability|countable]] [[union]] of nowhere dense sets: any other topological space is of '''second category'''. The ''[[Baire category theorem]]'' states that a non-empty [[complete metric space]] is of second category. | ||
Revision as of 16:01, 3 January 2009
In general topology, a nowhere dense set in a topological space is a set whose closure has empty interior.
An infinite Cartesian product of non-empty non-compact spaces has the property that every compact subset is nowhere dense.
A finite union of nowhere dense sets is again nowhere dense.
A first category space or meagre space is a countable union of nowhere dense sets: any other topological space is of second category. The Baire category theorem states that a non-empty complete metric space is of second category.
References
- J.L. Kelley (1955). General topology. van Nostrand, 145.