2
$\begingroup$

Here is an exercise from Abbott, Understanding Analysis:

enter image description here

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.

$\endgroup$
  • 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
3
$\begingroup$

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".

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.