A Klein surface is, as for Riemann surfaces, a surface with an atlas allowing the transition functions to be composed using complex conjugation. One can obtain the so called dianalytic structure of the space.
Read more about this topic: Klein Bottle
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“Voluptuaries, consumed by their senses, always begin by flinging themselves with a great display of frenzy into an abyss. But they survive, they come to the surface again. And they develop a routine of the abyss: “It’s four o’clock ... At five I have my abyss.””
—Colette [Sidonie Gabrielle Colette] (1873–1954)