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I'm reading "L'hypothèse du continu" by Sierpinski. He mentions many times "ensembles linéaires" or "linear sets" without defining this notion. Does anyone know what the definition of a such a set is ?

Here is a translation into English of 2 propositions he gave :

"There exists a linear set whose cardinality is that of the continuum and whose image by any continuous function is of measure zero".

"There exists a linear set whose cardinality is that of the continuum and whose homeomorphic images have the same measure".

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2 Answers 2

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After checking some translations, I can safely say that by ensemble linéaire he means a subset of the continuum, i.e., of $\Bbb R$.

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A set is said to be a linear set if one of the following conditions is satisfied:

  • it is empty;
  • it contains single element;
  • it satisfies the following condition:

    There are unique elements called "first" and "last" such that every element except the last has a unique successor, and every element except the first has a unique predecessor.

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  • $\begingroup$ Your answer differs significantly from Brian M. Scott's. Are you sure you've paid due attention to the context in which the question arose? $\endgroup$
    – Lord_Farin
    Jun 6, 2013 at 9:37
  • $\begingroup$ According to this definition, the set $[0,1]\cup(2,3]$ would not be a linear set. $\endgroup$
    – M.G
    Jun 6, 2013 at 13:49
  • $\begingroup$ Also, real numbers do not have immediate successors. $\endgroup$
    – Moni145
    Jul 11, 2021 at 2:30

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