Can a 1-side, 1-border object exist in 3D?

Question

We are three friends discussing whether a three dimensional object with a single side and a single are can possibly exist.

1. I first came up with a Moebius strip as an affirmative example
2. The second friend came up with a Klein bottle with a hole / drill (in order to provide the border)
3. The third and last friend argues that neither the strip or the bottle are valid examples because a Klein´s bottle is really a projection of a R4 object in R3, and because, in any case, both objects must have thickness in order to exist, so the strip would actually have two borders instead of just one.

Can a 1-side, 1-border object exist in 3D? Or is the third friend right with his "must have thickness in R3 in order to <> (be fisically feasible)"

Many Thanks

Answer 1

The Moebius strip is an obvious affirmative example, it has one border and one side. It proves that the answer to your question is yes.

The Klein bottle with a hole is a moebius strip. And your third friend is entirely wrong, a klein bottle with a hole can be embedded into $\mathbb R^3$.