---
tag: 04AJE0M271
kind: example
name: Möbius strip
aliases: [mobius-band]
msc: [57K20]
instance_of: [0CQHFPGJ45]
uses: [0V8046R0P9]
sources:
  - url: https://en.wikipedia.org/wiki/M%C3%B6bius_strip
    locator: lead section
  - url: https://en.wikipedia.org/wiki/M%C3%B6bius_strip
    locator: Properties
same_as:
  wikidata: Q226843
  wikipedia: 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|0V8046R0P9]], 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.