Matches in DBpedia 2016-04 for { <http://dbpedia.org/resource/Hausdorff_paradox> ?p ?o }
Showing triples 1 to 42 of
42
with 100 triples per page.
- Hausdorff_paradox abstract "The Hausdorff paradox is a paradox in mathematics named after Felix Hausdorff. It involves the sphere S2 (a 2-dimensional sphere in R3). It states that if a certain countable subset is removed from S2, then the remainder can be divided into three disjoint subsets A, B and C such that A, B, C and B ∪ C are all congruent. In particular, it follows that on S2 there is no finitely additive measure defined on all subsets such that the measure of congruent sets is equal (because this would imply that the measure of B ∪ C is simultaneously 1/3 and 2/3 of the non-zero measure of the whole sphere).The paradox was published in Mathematische Annalen in 1914 and also in Hausdorff's book, Grundzüge der Mengenlehre, the same year. The proof of the much more famous Banach–Tarski paradox uses Hausdorff's ideas.This paradox shows that there is no finitely additive measure on a sphere defined on all subsets which is equal on congruent pieces. (Hausdorff first showed in the same paper the easier result that there is no countably additive measure defined on all subsets.) The structure of the group of rotations on the sphere plays a crucial role here — the statement is not true on the plane or the line. In fact, as was later shown by Banach, it is possible to define an \"area\" for all bounded subsets in the Euclidean plane (as well as \"length\" on the real line) in such a way that congruent sets will have equal \"area\". (This Banach measure, however, is only finitely additive, so it is not a measure in the full sense, but it equals the Lebesgue measure on sets for which the latter exists.) This implies that if two open subsets of the plane (or the real line) are equi-decomposable then they have equal area.".
- Hausdorff_paradox wikiPageExternalLink Hausdorff_Paradox.
- Hausdorff_paradox wikiPageID "634759".
- Hausdorff_paradox wikiPageLength "2998".
- Hausdorff_paradox wikiPageOutDegree "19".
- Hausdorff_paradox wikiPageRevisionID "697026846".
- Hausdorff_paradox wikiPageWikiLink Banach_measure.
- Hausdorff_paradox wikiPageWikiLink Banach–Tarski_paradox.
- Hausdorff_paradox wikiPageWikiLink Category:Mathematics_paradoxes.
- Hausdorff_paradox wikiPageWikiLink Category:Measure_theory.
- Hausdorff_paradox wikiPageWikiLink Category:Theorems_in_analysis.
- Hausdorff_paradox wikiPageWikiLink Congruence_(geometry).
- Hausdorff_paradox wikiPageWikiLink Countable_set.
- Hausdorff_paradox wikiPageWikiLink Felix_Hausdorff.
- Hausdorff_paradox wikiPageWikiLink Grundzüge_der_Mengenlehre.
- Hausdorff_paradox wikiPageWikiLink Lebesgue_measure.
- Hausdorff_paradox wikiPageWikiLink Mathematics.
- Hausdorff_paradox wikiPageWikiLink Mathematische_Annalen.
- Hausdorff_paradox wikiPageWikiLink Measure_(mathematics).
- Hausdorff_paradox wikiPageWikiLink Rotation_group_SO(3).
- Hausdorff_paradox wikiPageWikiLink Sphere.
- Hausdorff_paradox wikiPageWikiLink Stefan_Banach.
- Hausdorff_paradox wikiPageWikiLinkText "Hausdorff paradox".
- Hausdorff_paradox wikiPageUsesTemplate Template:Cite_journal.
- Hausdorff_paradox subject Category:Mathematics_paradoxes.
- Hausdorff_paradox subject Category:Measure_theory.
- Hausdorff_paradox subject Category:Theorems_in_analysis.
- Hausdorff_paradox hypernym Paradox.
- Hausdorff_paradox type Redirect.
- Hausdorff_paradox type Theorem.
- Hausdorff_paradox comment "The Hausdorff paradox is a paradox in mathematics named after Felix Hausdorff. It involves the sphere S2 (a 2-dimensional sphere in R3). It states that if a certain countable subset is removed from S2, then the remainder can be divided into three disjoint subsets A, B and C such that A, B, C and B ∪ C are all congruent.".
- Hausdorff_paradox label "Hausdorff paradox".
- Hausdorff_paradox sameAs Q1959890.
- Hausdorff_paradox sameAs ハウスドルフのパラドックス.
- Hausdorff_paradox sameAs Hausdorff-paradox.
- Hausdorff_paradox sameAs Paradòss_ëd_Hausdorff.
- Hausdorff_paradox sameAs m.02z00n.
- Hausdorff_paradox sameAs Теорема_Хаусдорфа.
- Hausdorff_paradox sameAs Парадокс_Хаусдорфа.
- Hausdorff_paradox sameAs Q1959890.
- Hausdorff_paradox wasDerivedFrom Hausdorff_paradox?oldid=697026846.
- Hausdorff_paradox isPrimaryTopicOf Hausdorff_paradox.