Characterization of order types of pointwise linearly ordered families of Baire class 1 functions

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• 作者：M&aacute ; rton Elekes^a ; ^b ; ^{elekes.marton@renyi.mta.hu} ; Zolt&aacute ; n Vidny&aacute ; nszky^a ; ¹ ; ^{vidnyanszky.zoltan@renyi.mta.hu}
• 关键词：primary ; 26A21 ; 03E15 ; secondary ; 03E04 ; 03E50
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：5 February 2017
• 年：2017
• 卷：307
• 期：Complete
• 页码：559-597
• 全文大小：648 K
• 卷排序：307

文摘

In the 1970s M. Laczkovich posed the following problem: Let B1(X)B1(X) denote the set of Baire class 1 functions defined on an uncountable Polish space X equipped with the pointwise ordering.Characterize the order types of the linearly orderedsubsets of B1(X). The main result of the present paper is a complete solution to this problem.We prove that a linear order is isomorphic to a linearly ordered family of Baire class 1 functions iff it is isomorphic to a subset of the following linear order that we call ([0,1]↘0<ω1,<altlex), where [0,1]↘0<ω1 is the set of strictly decreasing transfinite sequences of reals in [0,1][0,1] with last element 0, and <altlex<altlex, the so called alternating lexicographical ordering  , is defined as follows: if (xα)α≤ξ,(xα′)α≤ξ′∈[0,1]↘0<ω1 are distinct, and δ is the minimal ordinal where the two sequences differ then we say that(xα)α≤ξ<altlex(xα′)α≤ξ′⇔(δ is even and xδ<xδ′) or (δ is odd and xδ>xδ′).Using this characterization we easily reprove all the known results and answer all the known open questions of the topic.