Klein bottle

Example

Let $Q = [0, 1] \times [0, 1]$ be the unit square. The Klein bottle $K$ is the quotient of $Q$ by the identifications $$(0, y) \sim (1, y) \quad \text{and} \quad (x, 0) \sim (1 - x, 1) \qquad \text{for } x, y \in [0, 1].$$ Then $K$ is a closed surface. Check: every point has a disc neighbourhood, made of pieces of $Q$ glued along the identified edges. Also $K$ is compact and connected as a quotient of $Q$. The four corners become one vertex, the edges give two edges, and $Q$ gives one face, so the Euler characteristic is $\chi(K) = 1 - 2 + 1 = 0$. The strip $\set{(x, y) : \abs{x - 1/2} \le 1/4}$ with its top and bottom edges glued by $(x, 0) \sim (1 - x, 1)$ is a Möbius strip in $K$, so $K$ is not orientable.