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 $$\scriptstyle D^2 \times I$$ 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 $$\scriptstyle M\ddot{o}\times I$$, of the möbius strip and an interval $$\scriptstyle I=[0,1]$$. 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: $$\scriptstyle M\ddot{o}\times[\frac{1}{2}-\varepsilon,\frac{1}{2}+\varepsilon]$$ and whose boundary is a Klein bottle.