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 . ${\displaystyle (X,{\mathcal {O}})}$

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 ${\displaystyle G\subset X}$ ${\displaystyle X}$ ${\displaystyle O_{n}\in {\mathcal {O}}}$ ${\displaystyle n\in \mathbb {N} }$

${\displaystyle G=\bigcap _{n\in \mathbb {N} }O_{n}}$

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. ${\displaystyle F\subset X}$ ${\displaystyle X}$

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

1. ↑ Eric W. Weisstein : F-Sigma Set . In: MathWorld (English).
2. ↑ Elstrodt: Measure and Integration Theory. 2009, p. 26.
3. ↑ Eric W. Weisstein : G-Delta Set . In: MathWorld (English).