Hausdorff's G _δ theorem

The -Satz of Hausdorff is a sentence from the mathematical branch of topology . It was published in the Fundamenta Mathematicae by Felix Hausdorff in 1924 . The -Satz is also the Russian mathematician by some authors Paul Alexandroff attributed that the sentence for the separable case for the special case, that is Polish areas had proved. ${\displaystyle G_{\delta }}$ ${\displaystyle G_{\delta }}$

Formulation of the sentence

In a complete metric space , a subspace that is a set , i.e. the intersection of countably many open subsets , can always be completely metrized . ${\displaystyle G_{\delta }}$

reversal

The -theorem has a certain inversion in the Mazurkiewicz theorem, which was proven by Stefan Mazurkiewicz : ${\displaystyle G_{\delta }}$

In a metric space , every fully metrizable subspace is a -set. ${\displaystyle G_{\delta }}$

Inference

From the -theorem it follows immediately that the set of irrational numbers ${\displaystyle G_{\delta }}$

${\displaystyle \mathbb {R} \setminus \mathbb {Q} =\bigcap _{q\in \mathbb {Q} }({\mathbb {R} \setminus \{q\}})}$

with that of deriving subspace topology is completely metrizable. ${\displaystyle \mathbb {R} }$This can be shown constructively by specifying a homeomorphism to the Baire space .

literature

Original work

• Paul Alexandroff : Sur les ensembles de la première classe et les ensembles abstraits . In: Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences . Vol. 178, 1924, pp. 185-187. Digitized .
• Felix Hausdorff : The sets G _δ in complete spaces . In: Fundamenta Mathematicae . Vol. 6, 1924, pp. 146-148. Digitized version (PDF; 129 kB) .
• Stefan Mazurkiewicz : About Borel Sets. In: Bulletin International de l'Académie des Sciences de Cracovie, Classe des Sciences Mathématiques et Naturelles. Series A: Sciences Mathématiques. 1916, ZDB -ID 761846-3 , pp. 490-494.

Books

• Felix Hausdorff: Collected Works. Including the philosophical and literary writings published under the pseudonym Paul Mongré and selected texts from the estate. Volume 3: Set theory (1927, 1935), descriptive set theory and topology. Edited by Ulrich Feigner, Horst Herrlich, Mirek Hušek, Vladimir Kanovei, Peter Koepke, Gerhard Preuß , Walter Purkert and Erhard Scholz . Springer, Berlin et al. 2008, ISBN 978-3-540-76806-7 .
• Stephen Willard : General Topology. Addison-Wesley, Reading MA et al. 1970.

Individual evidence

1. ↑ Vladimir Kanovei, Walter Purkert: Set theory - historical introduction. In: Hausdorff: Collected works. Volume 3. 2008, pp. 1-40, here p. 17; Hausdorff: The sets G _δ in complete spaces. In: Hausdorff: Collected works. Volume 3. 2008, pp. 443-453, here pp. 445-447.
2. ↑ Hausdorff: The sets G _δ in complete spaces. 1924, pp. 146-148.
3. ^ About Willard: General Topology. 1970, pp. 179, 310.
4. ↑ Hausdorff: The sets G _δ in complete spaces. 1924, pp. 146–148, here p. 146.
5. ^ Mazurkiewicz: On Borel sets. 1916, pp. 490-494.
6. ^ Willard: General Topology. 1970, pp. 179, 310-311.
7. ^ Willard: General Topology. 1970, p. 182.