Nowhere dense set: Difference between revisions
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In [[general topology]], a '''nowhere dense set''' in a topological space is a set whose [[closure ( | {{subpages}} | ||
In [[general topology]], a '''nowhere dense set''' in a topological space is a set whose [[closure (topology)|closure]] has empty [[interior (topology)|interior]]. | |||
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. | ||
==References== | ==References== | ||
* {{cite book | author=J.L. Kelley | authorlink=John L. Kelley | title=General topology | publisher=van Nostrand | year= 1955 | pages=145 }} | * {{cite book | author=J.L. Kelley | authorlink=John L. Kelley | title=General topology | publisher=van Nostrand | year= 1955 | pages=145,201 }} | ||
{{reflist}}[[Category:Suggestion Bot Tag]] | |||
Latest revision as of 06:01, 27 September 2024
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, 145,201.