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The sets of the form of all $(a, b]$ intervals in $(0, 1]$ is given to be a semialgebra.

If you take an inf intersection of $(\frac{a -1}{n}, b]$, for $n \to \infty$, for some valid $a$ and $b$ in $(0,1]$ you'll get the closed set $[a, b]$.

This is therefore an element but its complement is $(0,a) \cup (b, 1]$ but $(0,a)$ is not an element of the semialgebra hence its complement is not a union of disjoint elements of the semialgebra.

Sorry if I'm being stupid, I just don't know where I'm going wrong.

Your help would be appreciated,

Thanks

Willie Wong
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Dilan
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1 Answers1

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The complement of $(a,b]$ is $(0,a]\cup (b,1]$.

$[a,b]$ is not in your set, because semialgebras are closed under finite intersections, not arbitrary intersections. Your set is only half-open intervals.

vadim123
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