Explanation of non-orientability of the Möbius band

Question

I have read about the orientation of manifold in the Tu's book. The book is very readable but the first example about non-orientable manifold is seemly hard to understand. On page 208, he gave an argument that Möbius band is not orientable. Can anybody help me by some instructions?

Thanks.

Answer 1

Try reading some other discussions (http://en.wikipedia.org/wiki/Orientability, http://en.wikipedia.org/wiki/M%C3%B6bius_strip) and then return to the book you are reading. Making a physical model is also a very good idea. Or just look at a cool picture:

http://alpha.zimage.com/~ant/antfarm/ants/moebius-ants.jpg

Answer 2

Make a Mobius band from a strip of paper: glue the ends together with a half-twist.

Try to color one side red and the "other side" blue with crayons. You will find that you cannot distinguish one side from the other globally.

Seriously.

If you haven't tried this as a kid, you really should do this physical exercise now, it won't take long.

The Mobius strip is such a useful touchstone example it is worth it to understand it very well.