Solid Klein bottle explained

In mathematics, a solid Klein bottle is a three-dimensional topological space (a 3-manifold) whose boundary is the Klein bottle.^[1]

to the bottom disk by a reflection across a diameter of the disk. , of the möbius strip and an interval . In this model one can see that the core central curve at 1/2 has a regular neighbourhood which is again a trivial cartesian product: and whose boundary is a Klein bottle.

4D Visualization Through a Cylindrical Transformation

One approach to conceptualizing the solid klein bottle in four-dimensional space involves imagining a cylinder, which appears flat to a hypothetical four-dimensional observer. The cylinder possesses distinct "top" and "bottom" four-dimensional surfaces. By introducing a half-twist along the fourth dimension and subsequently merging the ends, the cylinder undergoes a transformation. While the total volume of the object remains unchanged, the resulting structure possesses a singular continuous four-dimensional surface, analogous to the way a Möbius strip has one continuous two-dimensional surface in three-dimensional space, and a regular 2d manifold klein bottle as the boundry.

Notes and References

1. .