TITLE: Klein-bottle and Möbius-strip together with a homeomorphism
QUESTION [12 upvotes]: Consider the Klein bottle (this can be done by making a quotient space). I want to give a proof of the following statement:
The Klein Bottle is homeomorphic to the union of two copies of a Möbius strip joined by a homeomorphism along their boundaries.
I know what such a Möbius band looks like and how we can obtain this also by a quotient map. I also know how to see the Klein bottle, but I don't understand that the given statement is correct. How do you construct such a homeomorphism explicitly?

REPLY [13 votes]: Let's directly glue two Möbius strips into a Klein bottle.

Take a Möbius strip and make its bottom wider:



Make the rear part of the band like a half of a bottle, also the front part of the band like a tube.



Take another Möbius strip and repeat steps 1 & 2 for the new one. Then glue new strip with the old one along their boundaries. Now we get a Klein bottle.  


Therefore, a Klein bottle is homeomorphic to the glued Möbius strips.