Solid Klein bottle

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

It is homeomorphic to the quotient space obtained by gluing the top disk of a cylinder ${\displaystyle \scriptstyle D^{2}\times I}$ to the bottom disk by a reflection across a diameter of the disk.

Alternatively, one can visualize the solid Klein bottle as the trivial product ${\displaystyle \scriptstyle M{\ddot {o}}\times I}$, of the möbius strip and an interval ${\displaystyle \scriptstyle I=[0,1]}$. 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: ${\displaystyle \scriptstyle M{\ddot {o}}\times [{\frac {1}{2}}-\varepsilon ,{\frac {1}{2}}+\varepsilon ]}$ 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" two-dimensional surfaces. By introducing a half-twist along the fourth dimension and subsequently connecting the ends, the cylinder undergoes a transformation. While the total volume of the object remains unchanged, the resulting structure is a continuous three-dimensional manifold - analogous to the way a Möbius strip is one continuous two-dimensional surface in three-dimensional space - and has the usual 2 dimensional Klein bottle manifold as its boundary.