As you move the point P around the surface of the Mobius strip, it reconnects to its original spot. But, given the same strip of paper laying on a table top, there would be two surfaces, a top and a bottom, that the point P would have to travel on.
Something to ponder: it seems we eliminated the 3rd dimension when we joined the two ends.
Isn’t that cool? This is just one illustration of how mathematical thinking can feed our natural curiosity and provide a desire to explore more mathematical concepts.