Can I safely say that the set of finite and cofinite subsets of the integers equipped with operations of union and intersection is isomorphic to the direct product of countably infinitely many $\mathbb Z_2$?
Set of finite and cofinite subsets of the integers
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number-theory
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2Definitely not: the direct product of countably infinitely many copies of $\mathbb{Z}_2$ is uncountable, but the set of finite and cofinite subsets of the integers is countable. Did you mean the direct sum? – 2012-02-16
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0@BrianM.Scott: Can the direct sum represent both the finite and cofinite subsets? – 2012-02-16
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1You can safely say that two structures are isomorphic when you have exhibited an isomorphism. – 2012-02-16
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0The direct product would correspond to the set of _all_ subsets, not just those that are finite or cofinite. – 2012-02-16
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0Quite possibly not $-$ I’ve not thought much about it $-$ but at least it has the right cardinality, unlike the direct product. However, it occurs to me that you have another fatal problem: neither union nor intersection is a group operation. You have identities for each, but no inverses. – 2012-02-16
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0I wonder whether OP is thinking of "symmetric difference" and intersection. The set is a ring under that pair of operations. – 2012-02-17