In an exercise it is asked to triangulate the Klein Bottle, and it is presented by this octagon. I really can't see a Klein Bottle here.
2026-03-25 20:41:34.1774471294
Is this really a Klein Bottle?
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As Lee Mosher points out, this fails to even be a surface. The reason is that a typical open neighborhood of a point in the interior of an "$a$" or "$b$" edge looks like three open half-disks glued together along their diameter. But this space cannot be homeomorphic to the plane. Indeed, removal of a point inside the common diameter yields a space homotopic to three intervals with corresponding endpoints identified, and this in turn is homotopic to a wedge of two circles. On the other hand, removing a point from inside the plane results in a space homotopic to a circle.