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I have these following sets (the overbar means closure) enter image description here

enter image description here

This doesn't work with open intervals such as $(a,b)$ or $(c,d)$ because at least two of these sets are equal. But I can't think of open subsets that satisfy this condition.

1 Answers 1

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Let $A=(0, 2)\cup(3,4)$ and $B=(1, 3)$. Then

$$A\cap \overline{B}=[1, 2)$$ $$\overline{A}\cap B=(1, 2]$$ $$\overline{A}\cap \overline{B}=[1, 2]\cup\{3\}$$ $$\overline{A\cap B}=[1, 2]$$

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    I wasn't sure about the closure of the union of open intervals. If the closure of $(a,b)$ is $[a,b]$ then the closure of $(a,b)\cap(c,d)$ is $[a,b]\cap[c,d]$? – 2017-02-01
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    @math4everyone Closure of the union is always equal to the union of closures. So for $\cup$ yes, but not for $\cap$. – 2017-02-01