* Re: limits of finite sets
@ 2004-07-31 12:02 Tom Leinster
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From: Tom Leinster @ 2004-07-31 12:02 UTC (permalink / raw)
To: categories
I asked for an explanation of the following result:
> the limit in Set of any diagram
>
> ... ---> S_3 ---> S_2 ---> S_1
>
> of finite nonempty sets is nonempty
Thanks very much to all who replied. I'll summarize some of the points
made to me in private replies:
1. This is called Koenig's Lemma, and is usually stated in the form
"any finitely-branching infinite tree contains an infinite
(positively oriented) path".
2. This also follows from a general result in topology by regarding
each S_n as a discrete space. The general result is that any
"suitably-shaped" limit of nonempty compact Hausdorff spaces is
nonempty. For Bourbaki (General Topology), "suitably-shaped" means
indexed by a directed poset. More generally still, it could be any
componentwise cofiltered limit, i.e. any limit for which each
connected-component of the indexing category I is cofiltered (or
equivalently, every finite connected diagram in I admits a cone).
The proof of the general topological result specializes to give a
nice topological proof of Koenig. For each n, let V_n be the
subset of the product \prod_n S_n consisting of those sequences
whose first n terms are compatible; then \lim_n S_n is the
intersection of the (V_n)s. But with the discrete topology on each
S_n, Tychonoff says that \prod_n S_n is compact, and (V_n) is a
nested sequence of nonempty closed subsets so has nonempty
intersection.
Tom
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