Il 12/05/2021 06:08, Timothy Y. Chow ha scritto:
> On Mon, 10 May 2021, I wrote:
>> The reason I ask is that I was recently refreshing my memory about
>> some of those results, and it seems that some version of the Vitali
>> covering lemma is typically used when lifting a map to a covering
>> space. So maybe the weak weak Koenig lemma is needed?
>
> I looked more carefully and found that in Munkres's textbook
> "Topology," the key lemma seems to be something he calls the "Lebesgue
> number lemma":
>
> Lemma. Let A be an open covering of a compact metric space. Then
> there exists delta > 0 such that every subset of X with diameter less
> than delta is contained in some element of A.
>
> Is this Lemma provable in RCA_0?
>
> Tim
The spaces satisfying the conclusion of the Lemma are called "Lebesgue
spaces" in the paper "Lebesgue numbers and Atsuji spaces in subsystems
of second-order arithmetic" by Giusto and myself (Arch. Math. Logic
(1998) 37: 343–362). In that paper we studied the strength of the lemma,
which depends on what we mean by "compact":
* RCA_0 proves that every Heine-Borel compact space (every countable
open cover has a finite subcover) is Lebesgue;
* WKL_0 is equivalent to the statement that every compact space (there
exists a uniform sequence witnessing totally boundedness) is Lebesgue.
That old paper studies several other implications, including the
property of being Atsuji (every continuous function with domain X is
uniformly continuous), which is equivalent to being Lebesgue. A couple
of questions were left open, and as far as I know are still so.
Best wishes,
Alberto
--
Alberto Marcone alberto.marcone at uniud.it
Dipartimento di Scienze Matematiche,
Informatiche e Fisiche
Universita' di Udine tel: +39-0432-558482
via delle Scienze 206 fax: +39-0432-558499
33100 Udine
Italy http://users.dimi.uniud.it/~alberto.marcone/
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