/sci/ - Science & Math » Thread #13999791

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Anonymous Thu 23 Dec 10:52:42 2021 No.13999791 View Reply Original Report

Quoted By: >>13999802 >>13999807 >>13999989

Why do you not need Axiom of Choice to define the fundamental group ?

You have to pick a point as it's a pointed space you're considering.

Anonymous

Anonymous Thu 23 Dec 2021 10:53:25 No.13999795 Report

Quoted By:

I thought that was Sarkozy

Anonymous

Anonymous Thu 23 Dec 2021 10:59:55 No.13999802 Report

Quoted By:

>>13999791
Haven't touched this in a while, but aren't you already given the pointed space, i.e. you're given the base point. So no need for AC.

Anonymous

Anonymous Thu 23 Dec 2021 11:03:00 No.13999807 Report

Quoted By: >>13999848 >>13999863

>>13999791
the base point is largely auxiliary anyway

Anonymous

Anonymous Thu 23 Dec 2021 11:22:37 No.13999848 Report

Quoted By:

>>13999807
Thanks, Wikipedia.

Anonymous

Anonymous Thu 23 Dec 2021 11:31:45 No.13999863 Report

Quoted By: >>13999909

>>13999807

still needed to define it

Anonymous

Anonymous Thu 23 Dec 2021 11:51:17 No.13999909 Report

Quoted By: >>13999935

>>13999863
given the base point doesn't actually matter, why would you need to invoke choice to have a well-defined fundamental group?

Anonymous

Anonymous Thu 23 Dec 2021 11:56:26 No.13999921 Report

Quoted By:

because the space might not be path connected

Anonymous

Anonymous Thu 23 Dec 2021 12:01:35 No.13999935 Report

Quoted By: >>13999989

>>13999909

why do you not come up with a definition that doesn't use the base point ?

Anonymous

Anonymous Thu 23 Dec 2021 12:22:19 No.13999989 Report

Quoted By: >>14000035

>>13999791
A pointed space is a pair (X,x) with x a point in X. No need for axiom of choice.
>>13999935
Because there already is such a definition. It's called the fundamental groupoid.

Anonymous

Anonymous Thu 23 Dec 2021 12:32:44 No.14000035 Report

Quoted By: >>14000075 >>14000235

>>13999989
>A pointed space is a pair (X,x) with x a point in X.

Which point ? You're using x as its name so you picked one point, you made a choice.

>>13999989
>Because there already is such a definition. It's called the fundamental groupoid.

Irrelevant. I'm talking about the fundamental group which requires picking a basepoint.

Anonymous

Anonymous Thu 23 Dec 2021 12:51:38 No.14000075 Report

Quoted By:

>>14000035
>fundamental group which requires picking a basepoint
And that has nothing to do whatsoever with axiom of choice.

Anonymous

Anonymous Thu 23 Dec 2021 13:43:07 No.14000235 Report

Quoted By:

>>14000035
>which requires picking a basepoint.
why don't you actually explain why this should require choice?

Anonymous

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Anonymous Thu 23 Dec 2021 13:48:34 No.14000260 Report

Quoted By:

There's no "the fundamental group" of a space. There is only the fundamental group of a pointed space, so the base point is already given. In some cases all these will be isomorphic, sometimes not (disjoint union of a point and a circle).

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