Möbius strip

Example

Let $\sim$ be the equivalence relation on the square $[0, 1] \times [0, 1]$ generated by $(0, y) \sim (1, 1 - y)$ for all $y \in [0, 1]$. The quotient space $[0, 1]^2 / {\sim}$, with the quotient topology, is the Möbius strip.

Remarks

The gluing has a half-twist: the bottom of the left edge is glued to the top of the right edge. The Möbius strip is a non-orientable surface. Its boundary is one circle, not two. Without the twist, the gluing $(0, y) \sim (1, y)$ gives a cylinder, which is not homeomorphic to the Möbius strip.

Explanations

L1 human

Take a strip of paper, give one end a half-turn, and glue the two ends together. The result has only one side: an ant can walk to every point of the strip without crossing an edge.