G δ and F σ sets
In mathematics, special sets in topological spaces are called G δ -sets and F σ -sets . They play an important role in measure theory and also appear in the formulation of permanent properties of certain classes of topological spaces.
definition
A topological space is given .
A set is called a G δ -set if it is the countable intersection of open sets in . That is, there are sets for all such that
applies.
A set is called an F σ -set if it is the countable union of closed sets in . Equivalent to this is that the complement of the set is a G δ -set.
designation
The naming is explained as follows:
- F stands for fermé , French for closed, the σ for somme , French for sum and derived from this the union, similar to σ-additivity or σ-finiteness .
- G stands for area , since Felix Hausdorff called open sets areas, the δ for average.
use
The G δ -sets are named after the G δ -set by Hausdorff , for example , and they also play a central role in the closely related Mazurkiewicz theorem .
In addition, both G δ -sets and F σ -sets are always Borel sets and are in the second level of the Borel hierarchy .
Web links
- G-delta . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
- F sigma . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
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 .
- Jürgen Elstrodt : Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , doi : 10.1007 / 978-3-540-89728-6 .
Individual evidence
- ↑ Eric W. Weisstein : F-Sigma Set . In: MathWorld (English).
- ↑ Elstrodt: Measure and Integration Theory. 2009, p. 26.
- ↑ Eric W. Weisstein : G-Delta Set . In: MathWorld (English).