/sci/ - Science & Math » Thread #12847134

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Anonymous Wed 17 Mar 21:23:18 2021 No.12847134 View Reply Original Report

Quoted By: >>12847186

Couldn’t you say the limit of a sequence of any topological space is the set itself since it’s in the topology?

Anonymous

Anonymous Wed 17 Mar 2021 21:25:23 No.12847142 Report

Quoted By: >>12847148

No.

Anonymous

Anonymous Wed 17 Mar 2021 21:27:36 No.12847148 Report

Quoted By: >>12847171

>>12847142
Why not?

Given the trivial topology could you not say the set itself is the limit of all sequences set in the space?

Anonymous

Anonymous Wed 17 Mar 2021 21:33:20 No.12847161 Report

Quoted By:

A limit of a sequence is a point anon. Not every point in a sequence is a limit of that sequence. In general, none of them are.

Anonymous

Anonymous Wed 17 Mar 2021 21:35:56 No.12847171 Report

Quoted By: >>12847228

>>12847148
If you give $\mathbb{R}$ the trivial topology. Then every sequence has every element of $\mathbb{R}$ as a limit. Limits in general are not unique.
But the limits are still elements of $\mathbb{R}$ . Sets of real numbers like $\mathbb{R}$ itself are not limits.

Anonymous

Anonymous Wed 17 Mar 2021 21:39:11 No.12847186 Report

Quoted By: >>12849372

>>12847134
If you want to generalize the notion of convergence for an arbitrary category, you might have a look at the notion of filter.

https://ncatlab.org/nlab/show/filter

Since Top has cartesian product and coproducts, it makes sense to define filters on it.

Anonymous

Anonymous Wed 17 Mar 2021 21:49:22 No.12847228 Report

Quoted By: >>12849372

>>12847171
Makes sense thanks anon, I have a other source backing up what you said too (a set with a trivial topology has the discrete elements of the set as the limits of the sequence)

Anonymous

Anonymous Thu 18 Mar 2021 10:56:21 No.12849372 Report

Quoted By:

>>12847228
like >>12847186
says learn topology in toposes, when you do this you have to use locales in order to get the usual theorems from classical maths with topological spaces

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