If such a cloth exists, then one should no more worry about the orientation of our clothes, which troubled me sometimes. :P
Thus I am wondering
Does there exist a non-orientable surface with $3$ holes in $\mathbb R^3?$
Maybe this is not precisely equivalent with the imagination, but intuitively this is what one thinks at first.
And I know that the Möbius strip has one hole.
In any case, every help is appreciated. Thanks in advance.
People make mobius scarves. Also since our surface area is orientable, then if we were to wear a nonorientable surface, there would have to be some kind of nonsmoothness involved (ie, some kind of fold against your skin), which would be annoying as hell.