Klein bottle

Overview and basic construction

Klein bottle

The Klein bottle is the quotient space $[0,1]\times [0,1]/\sim$, or the unit square modulo the relations $(0,t)\sim (1,t)(t,0)\sim (1-t,1).$

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In algebraic topology

• Group actions on topological spaces: The Klein bottle is homeomorphic to a group of transformations acting on $\mathbb{R}$.
• Calculating fundamental groups: By the previous homeomorphism, the Klein bottle gives an example of a upshot a fundamental group group that is not abelian.