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SMOOTH PEANO FUNCTIONS FOR PERFECT SUBSETS OF THE REAL LINE

机译:实线完美子集的平滑PEANO函数

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摘要

In this paper we investigate for which closed subsets P of the real line R there exists a continuous map from P onto P~2 and, if such a function exists, how smooth can it be. We show that there exists an infinitely many times differentiable function f:R→R~2 which maps an unbounded perfect set P onto P~2. At the same time, no continuously differentiable function f:R→R~2 can map a compact perfect set onto its square. Finally, we show that a disconnected compact perfect set P admits a continuous function from P onto P~2 if, and only if, P has uncountably many connected components.
机译:在本文中,我们研究了对于实线R的哪些闭合子集P,存在从P到P〜2的连续映射,如果存在这样的函数,其平滑度如何。我们证明存在无限次可微函数f:R→R〜2,它将无界完美集P映射到P〜2上。同时,没有连续可微函数f:R→R〜2可以将紧凑的理想集合映射到其平方上。最后,我们证明,当且仅当P具有不可数的连接分量时,一个不连续的紧致完美集P允许从P到P〜2的连续函数。

著录项

  • 来源
    《Real analysis exchange》 |2014年第1期|57-72|共16页
  • 作者单位

    Department of Mathematics, West Virginia University, Morgantown, WV 26506-6310 and Department of Radiology, MIPG, University of Pennsylvania, Blockley Hall - 4th Floor, 423 Guardian Drive, Philadelphia, PA 19104-6021;

    Department of Mathematics, University of Scranton, Scranton, PA 18510-4666;

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  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

    Peano curve; space filing curve; differentiability;

    机译:豌豆曲线;空间填充曲线;差异性;

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