Solid Klein bottle

{{Short description|Three-dimensional topological space}}

In mathematics, a solid Klein bottle is a three-dimensional topological space (a 3-manifold) whose boundary is the Klein bottle.{{citation|title=How Surfaces Intersect in Space: An Introduction to Topology|volume=2|series=K & E series on knots and everything|first=J. Scott|last=Carter|publisher=World Scientific|year=1995|isbn=9789810220662|page=169|url=https://books.google.com/books?id=jPPiur0Dd6UC&pg=PA169}}.

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

File:Moxi003.JPG 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 $\backslash scriptstyle\; M\backslash ddot\{o\}\backslash times\; I$, of the möbius strip and an interval $\backslash scriptstyle\; I=[0,1]$. 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: $\backslash scriptstyle\; M\backslash ddot\{o\}\backslash times[\backslash frac\{1\}\{2\}-\backslash varepsilon,\backslash frac\{1\}\{2\}+\backslash varepsilon]$ 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 a regular 2d manifold klein bottle as its boundary.