mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a solid Klein bottle is a three-dimensional

topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

3-manifold In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane (geometry), plane (a tangent ...

) whose boundary is the

Klein bottle In mathematics, the Klein bottle () is an example of a Orientability, non-orientable Surface (topology), surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the ...

.. It is

homeomorphic In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...

to the quotient space obtained by gluing the top disk of a

cylinder A cylinder () has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infinite ...

\scriptstyle D^2 \times I

to the bottom disk by a reflection across a diameter of the disk. Moxi003

Alternatively, one can visualize the solid Klein bottle as the trivial product

\scriptstyle M\ddot\times I

, of the

möbius strip In mathematics, a Möbius strip, Möbius band, or Möbius loop is a Surface (topology), surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Bened ...

and an interval

\scriptstyle I=,1 /math>. 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 In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...

\scriptstyle M\ddot\times frac-\varepsilon,\frac+\varepsilon /math> 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 Four-dimensional space (4D) is the mathematical extension of the concept of three-dimensional space (3D). Three-dimensional space is the simplest possible abstraction of the observation that one needs only three numbers, called ''dimensions'' ...

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.

References

3-manifolds {{topology-stub