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  1. Define the 4 types of interval subsets of the real numbers.
  2. Is the union of an arbitrary number of open intervals also an open interval?
  3. Is the intersection of an arbitrary number of open intervals also an open interval
  4. What happens if you switch the word open to close in these two questions?

$\textbf{Define the 4 types of interval subsets of the real numbers.}$ Does it mean (a,b), [a,b], (a,b], and [a,b)?

$\textbf{Is the union of an arbitrary number of open intervals also an open interval?}$ Yes

$\textbf{Proof:}$ Let $x \in \cup_{i \in I} A_i=A$. Then $s \in A_i$ for some i. Since this is open, x has an $\epsilon$-neighbourhood lying completely inside $A_i$ and this is also inside A.

$\textbf{Is the intersection of an arbitrary number of open intervals also an open interval?}$ No. For example $$\cup_{n=1}^\infty (\sqrt{2}-\frac{1}{N}, \sqrt{2}+\frac{1}{N})=\{\sqrt{2}\}$$ is closed but not open.

$\textbf{What happens if you switch the word open to close in these two questions?} (i.e. the following two questions)$

  • $\textbf{Is the union of an arbitrary number of closed intervals also an closed interval?}$ No

For example, if $I_n$ is the closed interval $$I_n=[\frac{1}{n},1-\frac{1}{n}]$$, then the union of the $I_n$ is an open interval $$\cup_{n=1}^\infty I_n=(0,1)$$

  • $\textbf{Is the intersection of an arbitrary number of closed intervals also an closed interval?}$ Yes

$\textbf{Proof:}$ If $\{F_i: i \in I\}$ is an arbitrary collection of closed sets, then $F_i^c$ is open. By the De Morgan's laws, we have $$(\cap_{i \in I} F_i)^c=\cup_{i \in I} F_i^c$$, which is open. Thus $\cap_{i \in I} F_i$ is closed.

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  • $\begingroup$ This doesn't appear to be a question. But it all looks right. $\endgroup$ – genisage Sep 15 '14 at 20:45
  • $\begingroup$ @genisage The question is in the first part. $\endgroup$ – Username Unknown Sep 15 '14 at 20:46
  • $\begingroup$ But you answered all of the questions from the first part. $\endgroup$ – genisage Sep 15 '14 at 20:49
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    $\begingroup$ Yes, that's what it means. Why include all of the other stuff? $\endgroup$ – genisage Sep 15 '14 at 21:01
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    $\begingroup$ Too many parts! The union of open intervals need not be an open interval. For example consider $(0,1)\cup (17,222)$. The union is an open set but not an open interval. $\endgroup$ – André Nicolas Sep 15 '14 at 21:08

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