Here is an exercise from Abbott, Understanding Analysis:

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I understand the construction of the Cantor set but I can't figure out what is meant here. Does he mean the closed or the open interval $(0,1)$? And which is the middle fourth, the left one or the right one?

More seriously: I'm interested in attempting to solve this exercise but I can't figure out what would make the most sense. Thanks for any clues.

  • 2
    $\begingroup$ I can't tell what the "open" should mean. But the middle fourth of $[0,1]$ is $(3/8,5/8)$. $\endgroup$ – Daniel Fischer Jan 3 '14 at 16:42
  • $\begingroup$ @Phira uh, isn't this the opposite of what the answer is saying? Any suggestions on how I can find out which is more likely intended by the author? $\endgroup$ – newb Jan 3 '14 at 17:38
  • $\begingroup$ @newb Yes, I am sorry for my mistake. You want to have closed sets at each step. $\endgroup$ – Phira Jan 3 '14 at 18:13

The middle fourth is an interval of length $1/4$ whose center is $1/2$.

If this is to be like the Cantor set, he must have meant the interval $[0,1]$, including the endpoints. If he calls it "open", maybe that could mean it's an open subset of the space $[0,1]$, but I would not call it an "open interval".


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