Does this count as a loop on Möbius strip

Question

Suppose we take the Möbius strip as $X = \frac{[0, 1]\times[0, 1]}{\sim}$ with usual equivalence relation.

If I define $\alpha: [0, 1] \rightarrow X$ by $x \rightarrow [(x, 1/2)]$, is this a loop? Because it seems to start and end at the same point, although that point is on the opposite side of the Möbius strip?

More generally, when defining a loop on a surface, does it matter if the start and end point are 'the same point' but on opposite sides of the surface?

Answer 1

What you have written is a loop. Indeed, it is a continuous map $\alpha:[0,1]\to X$ so that $\alpha(0)=\alpha(1)$. Regarding "sides" of surfaces, you should note that the Möbius strip $X$ has only one side in some sense, and this relates to the fact that it is nonorientable. In general, however, we do not worry about sides, as it is not really well-defined to say what side of a surface a curve is on in the first place.

For instance, take the equator of $S^2$. Which side of $S^2$ is this on?

Answer 2

In fact, being on the same face or not is not here a rigorous statement: a point on the mobius strip is a point on the mobius strip. It does not belong to any of its face.