---
tag: VGJJVBZ1MQ
kind: concept
name: Surface
msc: [57K20, 57N99]
specializes: [QQQ9B1DK4P]
sources:
  - url: https://en.wikipedia.org/wiki/Surface_(topology)
same_as:
  wikidata: Q484298
  wikipedia: Surface (topology)
---

## Summary

A surface is a [[topological manifold|QQQ9B1DK4P]] of dimension $2$: a space that looks like a piece of the plane near each of its points.
Spheres, tori, the Möbius strip and the Klein bottle are surfaces.
Closed surfaces are completely classified by orientability and Euler characteristic, see the [[classification of surfaces|2SFM73YET0]].
Surfaces are the first nontrivial case of manifold topology and appear throughout geometry, complex analysis and dynamics.

## Explanations

### L1 human

A surface is a space in which a tiny creature, looking around itself, would always believe it lives on a flat sheet.
The skin of a ball, the skin of a doughnut and an infinite plane are surfaces.
Two cones touching at their tips do not form a surface, because at the tips the creature would see two sheets meeting at a point.
Globally a surface can close up on itself, have holes, or even have no consistent inside and outside, like the Möbius strip.