Young's theorem (set theory)

The set of Young , named after William Henry Young , is a statement from the descriptive set theory and the theory of functions of a real variable, the amount of discontinuity describes a function.

With the help of Young's theorem and Baire's theorem, it can be shown, for example, that there can be no function that is discontinuous in all irrational places and continuous in all rational places.

Formulation of the sentence

The set of points of discontinuity of a function is a F _σ -set , i.e. a countable union of closed sets, while the set of points of continuity is a G _δ -set , i.e. a countable average of open sets. ${\displaystyle f\colon \mathbb {R} \rightarrow \mathbb {R} }$

One can also prove that for every -set there is a function such that the set is its points of discontinuity. ${\displaystyle F_{\sigma }}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {A}}}$

example

The function

${\displaystyle r(x)={\begin{cases}{\frac {1}{q}},&x\in \mathbb {Q} ,\ q=\min\{n:n\in \mathbb {N} ,\ nx\in \mathbb {Z} \}\\0,&x\in \mathbb {R} \setminus \mathbb {Q} ,\end{cases}}}$

which assigns the stem fraction with the same denominator to every rational number and maps irrational numbers to 0, is discontinuous at all rational places and continuous at all irrational places. As a countable union of closed (namely one-point) sets, the set of rational points is a -set: ${\displaystyle \mathbb {Q} }$ ${\displaystyle F_{\sigma }}$

${\displaystyle \mathbb {Q} =\bigcup _{x\in \mathbb {Q} }\left\{x\right\}}$

proof

See: Proof of Young's Theorem in the Evidence Archive

literature

• Felix Hausdorff : Fundamentals of set theory. Veit & Comp., Leipzig 1914 (Reprinted. Chelsea Publishing Company, New York NY 1949).
• Klaus Gloede: Script for the lecture Descriptive Set Theory . (PDF; 764 kB). Mathematical Institute of the University of Heidelberg, June 2006.