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| In [[mathematics]], a '''solid Klein bottle''' is a [[3-manifold]] (with boundary) [[homeomorphism|homeomorphic]] to the [[quotient space]] obtained by gluing the top of <math>\scriptstyle D^2 \times I</math> (cylinder) to the bottom by a reflection, i.e. the point <math>\scriptstyle (x,0)\,</math> is identified with <math>\scriptstyle (r(x), 1)\,</math> where <math>\scriptstyle r\,</math> is reflection of the disc <math>\scriptstyle D^2\,</math> across a diameter.
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| [[File:Moxi003.JPG|120px|thumb|right|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]]
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| Note the boundary of the solid Klein bottle is a [[Klein bottle]].
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| Alternatively, one can visualize the solid Klein bottle as the [[I-bundle|trivial product]] <math>\scriptstyle M\ddot{o}\times I</math>, of the [[möbius strip]] and an interval <math>\scriptstyle I=[0,1]</math>. In this model one can see that
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| the core central curve at 1/2 has a regular neighborhood which is again a trivial [[cartesian product]]: <math>\scriptstyle M\ddot{o}\times[\frac{1}{2}-\varepsilon,\frac{1}{2}+\varepsilon]</math> and whose boundary is a Klein bottle as previously noticed
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| [[Category:3-manifolds]]
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| {{topology-stub}}
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Latest revision as of 14:36, 26 November 2014
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