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Cantor set See the link, I am referring to cantor set on the real line. I wish to show that it is compact. I am doing this by pointing following arguments. I am not sure if this is enough.

  1. Cantor set is bounded by definition in the region $[0,1]$
  2. Cantor set is the union of closed intervals, and hence it is a closed set.
  3. Since the Cantor set is both bounded and closed it is compact by Heine-Borel Theorem.
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    $\begingroup$ 2. Complement of union of open intervals. $\endgroup$
    – user545497
    Mar 27 '18 at 3:50
  • $\begingroup$ Yes true. However is there something wrong with taking it as union of closed set. Both facts imply one another. $\endgroup$
    – Sonal_sqrt
    Mar 27 '18 at 3:55
  • $\begingroup$ A union of closed sets need not be closed, so you should amend statement 2 accordingly. $\endgroup$
    – Valborg
    Mar 27 '18 at 3:56
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    $\begingroup$ You could fix that by saying, each partial level of the construction is a finite union of closed intervals, and hence is closed, and the full set is the intersection of closed sets, and hence closed. $\endgroup$
    – Valborg
    Mar 27 '18 at 3:58
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    $\begingroup$ But the Cantor set is not a union of intervals of any kind: it’s totally disconnected, even. $\endgroup$
    – Lubin
    Mar 27 '18 at 4:26
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Cantor set is defined as $C=\cap_n C_n$ where $C_{n+1}$ is obtained from $C_n$ by dropping 'middle third' of each closed interval in $C_n$

As you have noted, Cantor set is bounded.

Since each $C_n$ is closed and $C$ is an intersection of such sets, $C$ is closed (arbitrary intersection of closed sets is a closed set).

As $C$ is closed and bounded, it is compact by Heine-Borel theorem.

PS: You cannot say that Cantor set is a union of closed intervals. Rudin is giving Cantor set as an example for a perfect set that contains no open interval!

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  • $\begingroup$ Thanks for the reply. However who is Rudin? $\endgroup$
    – Sonal_sqrt
    May 23 '18 at 10:20
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    $\begingroup$ It is Walter Rudin, his book "Principles of Mathematical Analysis" (nicknamed baby Rudin) is loved by many and hated by many. $\endgroup$ May 23 '18 at 10:22

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