Re: Cardinality question

"Bart Killam"
>
> In reviewing one of my old textbooks (Simmons, "Introduction to
> Topology and Modern Analysis", I got stuck on a simple problem in the
> first chapter. The hint he gives seems unrelated to the problem; I am
> stuck. The problem is 8.10 in the first chapter; the latex version of
> the problem is below.
>
> The ascii version is: given an infinite class of countable sets (Xi), i in
I,
> where I is the index set, show that the cardinality of Union(Xi), i in
> I, is less than or equal to the cardinality of the index set I.
>
> If I is countable, this is easy (the cardinalities are equal) ; if I
> is uncountable, the hint is to use Zorn's lemma to write Union(Xi),
> i in I, as the union of a disjoint class of countably infinite
> subsets.
>
> Zorn's lemma applies to a poset; it is not clear that the family
> (Xi), i in I is a poset with set inclusion as the order relation.
> If even I can show the hint, I do not see how writing the original
> union as a disjoint union solves the problem. As you can see, I am
> thoroughly confused. Any advice is appreciated.
This is not a trivial exercise. An equivalent statement is this:
If J is an infinite set and D is a denumerable infinite set, then
card(JxD) = card(J).
The hint is to apply this lemma, which can be proved
(not trivially) with the aid of Zorn's lemma:
Any infinite set can be partitioned into a family of countably
infinite sets.
Once you have the lemma, just partition J and use
card(NxN)=card(N).
I can spell it all out if needed.
LH


.

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