The Klein bottle and the projective plane are the basic non-orientable surfaces. The Klein bottle, obtained by gluing together two Mobius bands, is similar in some ways to the torus, and is something of a curiosity. The projective plane, obtained by gluing a disk to a Mobius band, is one of the most fundamental of all mathematical objects. Of all the surfaces, it most closely resembles the sphere.
This is the first video of the seventh lecture in this beginner's course on Algebraic Topology, given by Assoc Prof N J Wildberger of the School of Mathematics and Statistics at UNSW.
NOTE: This entire series is now available in the full (hour long lectures), also at this channel under the playlist AlgTop (full lectures):
http://www.youtube.com/playlist?list=PL6763F57A61FE6FE8&feature=view_all
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