This is a problem in " Basic set theory "- S.Shen: prove that if $ [0,1]=A \cup B$ then either $A$ or $B$ has the cardinality of the continuum. (Does this follow from the Cantor-Bernstein theorem?)
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2See this question. – bof Mar 03 '16 at 12:50
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@bof I thought this problem in a week but I counld'nt understand how does it work in closed unit interval? Could you explain for me? – Minh Nam Mar 10 '16 at 11:32
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@bof I can't understand what is horizontal line in the real line? – Minh Nam Mar 10 '16 at 11:48
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(1) Do you understand that there is a bijection between $[0,1]$ and $[0,1]\times[0,1]$? (2) Do you understand that, if $h:[0,1]\to[0,1]\times[0,1]$ is a bijection, and if $[0,1]=A\cup B,$ then $[0,1]\times[0,1]=h(A)\cup h(B)$? (3) Do you understand why it follows, as in the solution to that other question, that either $h(A)$ or $h(B)$ has the cardinality of the continuum? (4) And why it follows from that, that $A$ or $B$ has the cardinality of the continuum? – bof Mar 10 '16 at 12:04
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1thanks a lot @bof . I understand it now. – Minh Nam Mar 10 '16 at 12:06