# Transfinite donuts?

1. Apr 28, 2010

### 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?

2. Apr 28, 2010

### 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.

3. Apr 30, 2010

### lavinia

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

4. Apr 30, 2010

### 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?

Last edited: Apr 30, 2010
5. Apr 30, 2010

### lavinia

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.