Solid Klein bottle

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In mathematics, a solid Klein bottle is a 3-manifold (with boundary) homeomorphic to the quotient space obtained by gluing the top of \scriptstyle D^2 \times I (cylinder) to the bottom by a reflection, i.e. the point \scriptstyle (x,0)\, is identified with \scriptstyle (r(x), 1)\, where \scriptstyle r\, is reflection of the disc \scriptstyle D^2\, across a diameter.

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

Note the boundary of the solid Klein bottle is a Klein bottle.

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 as previously noticed