Solid Klein bottle

[1] It is homeomorphic to the quotient space obtained by gluing the top disk of a cylinder

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:

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.

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 boundary.