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.

Mö x I: the circle of black points marks an absolute deformation retract of this space, and any regular neighbourhood of it has again boundary as a Klein bottle, so Mö x I is an onion of Klein bottles

Alternatively, one can visualize the solid Klein bottle as the trivial product $\scriptstyle M\ddot{o}\times I$, 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 neighborhood 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.

References

1. Carter, J. Scott (1995), How Surfaces Intersect in Space: An Introduction to Topology, K & E series on knots and everything, 2, World Scientific, p. 169, ISBN 9789810220662.