I've got a conceptual question (that depends on definitions involved being unambiguous).

Is the Moebius strip as we tend to imagine it in $\displaystyle R^3$ a surface? (If so I would like to see the FORM of the intervals the parameter set)

Is it a compact surface if we imagine it with the edge curve? We imagine a compact set of $\displaystyle R^3$…but which definition of a compact surface, if any, is appropriate here?

Note that the Jordan Curve theorem allows to conclude that a compact surface in $\displaystyle R^3$ is orientable.

And remember that the Moebius band is non-orientable, although compact (!), as a 2-dim (quotient) manifold of a closed rectangle.

The question is essentially this: How does one not become a victim of a misconception?