Re: Cofinality

On Wed, 9 Apr 2008, David C. Ullrich wrote:
<marsh@xxxxxxxxxxxxxxxxxx> wrote:

Let S be a linear order.

Let A be a cofinal subset of S.

Is there a C subset A such that C is cofinal
to S and |C| = cof S, the cofinality of S?

Yes, at least assuming AC.

Say D is cofinal and has cardinality equal to the cofinality
of S. Since A is cofinal, for every d in D there exists a in A
with a > d. So there exists f : D -> A with f(d) > d for all d
in D. Let C = f(D). Then card(C) <= card(D), but since
A is cofinal C is also cofinal so card(C) >= card(D)
(since D has minimal cardinality among cofinal subsets).

Thanks David. With your approach I was able to prove the
equivalent, yet slicker theorem
C cofinal subset S implies cof C = cof S

The beauty of the proof is that the order need not be linear.

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