Solid Klein bottle

In mathematics, a solid Klein bottle is a three-dimensional topological space (a 3-manifold) whose boundary is the Klein bottle. It is homeomorphic to the quotient space obtained by gluing the top disk of a cylinder [math]\displaystyle{ \scriptstyle D^2 \times I }[/math] 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 [math]\displaystyle{ \scriptstyle M\ddot{o}\times I }[/math], of the möbius strip and an interval [math]\displaystyle{ \scriptstyle I=[0,1] }[/math]. 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: [math]\displaystyle{ \scriptstyle M\ddot{o}\times[\frac{1}{2}-\varepsilon,\frac{1}{2}+\varepsilon] }[/math] and whose boundary is a Klein bottle.