Surface (topology)
A surface is a two-dimensional topological manifold.
In topology, a surface is a two-dimensional manifold. Surfaces can arise as boundaries of three-dimensional solid figures, such as the sphere being the boundary of a solid ball, or as graphs of functions of two variables. They can also be defined abstractly without reference to any ambient space, as in the case of the Klein bottle, which cannot be embedded in three-dimensional Euclidean space. Topological surfaces are sometimes equipped with additional structures like Riemannian metrics or complex structures, connecting them to differential geometry and complex analysis.
- field
- Topology
- key_concept
- Two-dimensional manifold
- examples
- Sphere, torus, Klein bottle, real projective plane, Möbius strip
- properties
- Hausdorff, second-countable, locally Euclidean, often connected
- orientability
- Orientable (sphere, torus) or non-orientable (real projective plane, Möbius strip)
Lore & Background
A topological surface is defined as a topological space in which every point has an open neighborhood homeomorphic to some open subset of the Euclidean plane. This local Euclidean property allows the assignment of local coordinates. In most writings, a surface is assumed to be nonempty, second-countable, Hausdorff, and connected. Surfaces can be with or without boundary; a surface with boundary has points mapped to the x-axis of the upper half-plane, forming a one-dimensional boundary.
Reader's Guide
The concept of surface is fundamental in topology and geometry. Historically, surfaces were defined extrinsically as subspaces of Euclidean space, often as zero loci of functions. Modern intrinsic definitions treat the surface as a topological space without requiring an ambient space, though the Whitney embedding theorem shows every surface can be embedded in Euclidean space, specifically in E4. Compact surfaces without boundary are called closed surfaces; examples include the sphere, torus, and real projective plane. Orientability is a key property: a surface is orientable if it does not contain a homeomorphic copy of the Möbius strip. Surfaces can be constructed from polygons by identifying edges, as with fundamental polygons. The study of surfaces extends into differential geometry, algebraic geometry, and complex analysis, where additional structures like smoothness, metrics, or complex structures are added. Surfaces model physical objects and are used in physics, engineering, and computer graphics.
Did You Know?
- The Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space.
- The Möbius strip is a surface where clockwise and counterclockwise can be defined locally but not globally.
- Any compact surface that is either orientable or has a boundary can be embedded in E3.
- The real projective plane is compact, non-orientable, and without boundary, and cannot be embedded into E3.
More in Geometry And Shapes 1-24
Spotted an error? Know more?
This is a living reference — every entry is fact-audited, and reader corrections feed straight into our audit queue. Suggest an edit · See this site's audit record