Math of Transfinite Donuts in 3-D Space

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Discussion Overview

The discussion centers on the mathematical exploration of donuts, specifically two-dimensional closed surfaces with an infinite number of holes, situated in three-dimensional space. Participants inquire about existing mathematical frameworks or results related to these structures, including their properties and classifications.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions whether any mathematics has been developed regarding donuts with an infinite number of holes.
  • Another participant suggests that such structures might be as well understood as traditional donuts with a single hole, referencing Riemann surfaces as a well-established area of study.
  • A different participant proposes the possibility of constructing a donut with uncountably many holes, noting that it would not be a paracompact manifold.
  • One participant introduces a specific construction involving a unit disk with rational points deleted, questioning its classification as a torus.
  • Another participant expresses uncertainty about the toroidal nature of the previous example and suggests considering a Cartesian product of a circle with uncountably many holes removed.

Areas of Agreement / Disagreement

Participants express differing views on the classification and properties of donuts with infinite holes, indicating that multiple competing perspectives exist without a clear consensus.

Contextual Notes

Some claims depend on specific mathematical definitions and assumptions that are not fully articulated in the discussion, leaving certain aspects unresolved.

HarryWertM
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Has anyone ever developed any sort of math involving donuts with an infinite number of holes? By donut, I mean a two-dimensional closed surface, curved in 3-space, with one 'hole'. Are there any results, of any kind, for 2-D donuts in 3-D space, with infinite number of holes?
 
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I think they're about as well understood as the donut with one hole. Riemann surfaces are one of the most thoroughly understood branches of mathematics.
 
Could you construct a doughnut with uncountable many holes ? It would not be a paracompact manifold.
 
You mean something like S\times I, where S is the unit disk with the rational points inside a circle of radius 1/2 centered at the origin deleted?
 
Last edited:
Sine Nomine said:
You mean something like S\times I, where S is the unit disk with the rational points inside a circle of radius 1/2 centered at the origin deleted?

Not sure how that example is a torus.

I was thinking more of the long line Cartesian product the circle with uncountably many holes removed.
 

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