Math of Transfinite Donuts in 3-D Space

[HarryWertM]

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?

 

[zhentil]

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.

 

[lavinia]

Could you construct a doughnut with uncountable many holes ? It would not be a paracompact manifold.

 

[Sine Nomine]

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?

 

[lavinia]

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.