Nowhere dense crowd
Nowhere are dense sets in the set- theoretical topology special sets that are closely related to the dense sets , but not (as the name suggests) their opposite. For example, they form the basis for the formulation of Baire's theorem of categories , on which many far-reaching statements in functional analysis are based.
definition
A topological space is given . Then a lot means nowhere dense in when the inside of the close of is empty, so
- .
applies.
comment
The order of the conclusion and the interior are not interchangeable, as in general
is. For example, the real numbers are provided with the standard topology
- and thus ,
but if the operations are reversed it follows
- and thus .
Relation to dense sets
Dense sets and nowhere dense sets are related, but do not form a pair of opposites. So (everywhere) dense sets are not the complements of nowhere dense sets or those sets that are not nowhere dense. More precisely, a set is nowhere dense if and only if it is not dense in any non-empty open set (that is, dense in the corresponding subspace topology ). Every dense set is never dense anywhere, because it always follows from and the fact that the basic set of space is always open that and with it is. However, there are, for example , provided with the standard topology, both sets that are not dense and not dense anywhere (for example, the whole numbers ) and sets that are not dense and not dense anywhere like the interval .
Additional terms and usage
Sets that are nowhere a countable union of dense sets are called lean sets or sets of the first category. A crowd that is not lean is called a second-tier, or fat, crowd. Furthermore, the complement of a meager set is called komager.
Baire's theorem is based on these terms, which are based on the nowhere dense sets . This provides an abstract statement of existence and forms the basis for many far-reaching sentences of functional analysis .
literature
- Boto von Querenburg : Set theoretical topology . 3. Edition. Springer-Verlag, Berlin Heidelberg New York 2001, ISBN 978-3-540-67790-1 , doi : 10.1007 / 978-3-642-56860-2 .
- Hans Wilhelm Alt : Linear Functional Analysis . 6th edition. Springer-Verlag, Berlin Heidelberg 2012, ISBN 978-3-642-22260-3 , doi : 10.1007 / 978-3-642-22261-0 .
Web links
- Nowhere dense set . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
- Eric W. Weisstein : Nowhere Dense . In: MathWorld (English).