#Closure of Infinite Unions of Sets

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young raft
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I have a homework problem where we are asked to show that $$\overline{A}\cup\overline{B}=\overline{A\cup B}$$, which I proved with no difficulty. The next part of the question asks if this holds for an infinite union of sets. I want to say no, but I cannot find a counterexample. Can someone give me a hint without giving me the answer? I've tried things such as (a-1/n) where a is a natural number for example, but can't find one that doesn't work.

tacit hareBOT
molten wingBOT
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dingus

529059062420340737.png
tacit hareBOT
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idle belfry
young raft
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right. i couldnt think of an uncountably infinite number of sets tho

idle belfry
young raft
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i know that

idle belfry
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And are sets.

young raft
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so i just define a set of singletons where each singleton is a real number?

idle belfry
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The real numbers already are sets.

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Dedekind cuts.

young raft
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i have no clue what that means

idle belfry
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A Dedekind cut is a subset of the rational numbers that defines a real number. It has four properties; it is not empty, it is not equivalent to the set of rational numbers, it has no greatest element, and it contains all rational numbers smaller than any element it contains.

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That is, it's closed downward.

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Technically most descriptions of a Dedekind cut make it an ordered pair of two sets, the first being as I described and the second being its complement in the rationals, but the lower set is what counts.

young raft
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I don't think we can use that since we haven't learned it

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but i figured it out. thx for your help!

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+close

dire violetBOT
# young raft +close
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dire violetBOT
# dire violet

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