Category:Klein Bottle

This category contains results about the Klein bottle.
Definitions specific to this category can be found in Definitions/Klein Bottle.

A Klein bottle is a $2$-manifold obtained from the square:

$\set {\tuple {x_1, x_2} \in \R^2: \size {x_1}, \size {x_2} \le 1}$

by identifying the edges $x_1 = \pm 1$ and $x_2 = \pm 1$ such that:

$\tuple {-1, x_2}$ is identified with $\tuple {1, -x_2}$ for all $x_2$

$\tuple {x_1, -1}$ is identified with $\tuple {x_1, 1}$ for all $x_1$.

That is, by identifying both pairs of opposite sides, one with the other, of a square, while twisting the square for one pair: