What is a Klein bottle and its role in mathematics?

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A Klein bottle is a non-orientable surface with no distinct "inside" or "outside," similar to a Möbius strip, and is characterized by having only one continuous surface. Mathematically, it is expressed through topology and has an Euler characteristic of 0, classifying it among other complex surfaces. Its unique properties make it a subject of interest in various mathematical fields, including algebraic topology. Klein bottles are also explored in theoretical physics and computer graphics, illustrating their practical applications. Overall, the Klein bottle serves as an important concept in understanding higher-dimensional spaces and non-Euclidean geometry.
Gonzolo
I understand a Klein bottle is somewhat of a 3D version of a mobius strip in that it has only one continuous surface. But what do mathematicians do with it? How is it expressed mathematically, how does it classify among other shapes? Does it have some kind of applications?
 
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