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In the mathematical field of descriptive set theory, a set of real numbers (or more generally a subset of the Baire space or Cantor space) is called universally Baire if it has a certain strong regularity property. Universally Baire sets play an important role in Ω-logic, a very strong logical system invented by W. Hugh Woodin and the centerpiece of his argument against the continuum hypothesis of Georg Cantor. ==Definition== A subset ''A'' of the Baire space is universally Baire if it has one of the following equivalent properties: #For every notion of forcing, there are trees ''T'' and ''U'' such that ''A'' is the projection of the set of all branches through ''T'', and it is forced that the projections of the branches through ''T'' and the branches through ''U'' are complements of each other. #For every compact Hausdorff space Ω, and every continuous function ''f'' from Ω to the Baire space, the preimage of ''A'' under ''f'' has the property of Baire in Ω. #For every cardinal λ and every continuous function ''f'' from λω to the Baire space, the preimage of ''A'' under ''f'' has the property of Baire. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Universally Baire set」の詳細全文を読む スポンサード リンク
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