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My real analysis textbook says that the set $$[0,1] \cup (2,3]$$ has maximum and minimum, but it is not compact. At the same time, the Heine-Borel Theorem says that

A subset $S$ of $\mathbb R$ is compact iff $S$ is closed and bounded.

To my inexperienced eyes, the set $[0,1] \cup (2,3]$ is bounded, hence its maximum and minimum, and is also closed $-$ therefore according to the theorem it should be compact. Please let me know why I get so wrong $-$ thank you.

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    $\begingroup$ $[0,1]\cup(2,3]$ is not closed, why do you think it is? $\endgroup$ – SmileyCraft Dec 20 '18 at 18:12
  • $\begingroup$ That is exactly the reason I would like to get help. All I know is that a set is closed if its complement is open. Let me know how to show that it is not closed. Thank you. $\endgroup$ – Amanda.M Dec 20 '18 at 18:14
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    $\begingroup$ $2$ is in the complement, but for any ball of radius $\varepsilon>0$ around $2$, you can find an element not in the complement. Hence the complement is not open. $\endgroup$ – SmileyCraft Dec 20 '18 at 18:16
  • $\begingroup$ Thank you, that makes sense. $\endgroup$ – Amanda.M Dec 20 '18 at 18:25
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Yes, your set is bounded. But its not closed, since its complement is $(-\infty,0)\cup(1,2]\cup(3,+\infty)$, which is not open: $2$ belongs to it, but no open interval centered at $2$ is contained in it.

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  • $\begingroup$ Thank you also for adding "compactness" to my posting's category. $\endgroup$ – Amanda.M Dec 20 '18 at 18:31
  • $\begingroup$ I thought it was relevant. $\endgroup$ – José Carlos Santos Dec 20 '18 at 18:32
  • $\begingroup$ Do you think that henceforth intuitively, I may look at the set as union of two sets, $[0,1]$ as a closed set and $(2,3]$ as open set, and therefore the union is open? Thank you again. $\endgroup$ – Amanda.M Dec 20 '18 at 18:35
  • $\begingroup$ No. It is neither open nor closed. $\endgroup$ – dbx Dec 20 '18 at 18:36
  • $\begingroup$ Ok, I think I got it. But let me make my question a bit more precise: Do you think that henceforth intuitively, I may look at the problem as union of two sets, $[0,1]$ as a closed set and $(2,3]$ as open set, and therefore their union "can not be said as closed"? Thank you again. $\endgroup$ – Amanda.M Dec 20 '18 at 18:40

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