Math 2.0

Manifolds and cell complexes Simplicial complexes and piecewise-linear topology Theorem

Radó's theorem on triangulation of surfaces

Builds on Topological manifoldTriangulationSchoenflies theorem

Statement

Let SS be a topological manifold of dimension 22, that is, a Hausdorff, second-countable space locally homeomorphic to ℝ2\R^2. Then SS has a triangulation.

Remarks

Tibor Radó proved the theorem in 1925. One proof covers SS by countably many closed disks coming from charts, and modifies their boundary circles so that, on each compact part, they meet in finitely many points. The pieces cut out by these circles are bounded by Jordan curves, so the Schoenflies theorem shows that they are closed disks, which can then be divided into triangles. For example, the torus is triangulable, with as few as 77 vertices. The theorem is the reason why the classification of closed surfaces, first proved for triangulated surfaces, applies to all closed topological surfaces. The triangulation is also unique up to piecewise linear homeomorphism, so every surface has a unique piecewise linear structure. A related result, also called Radó's theorem, says that every connected Riemann surface is second-countable; its proof uses the same kind of construction.

Going deeper

For a student

In a course on surfaces, theorems are often proved first for triangulated surfaces, using the combinatorics of the triangles. Radó's theorem removes this hypothesis. The analogous statement holds in dimension 33 by Moise's theorem, but fails in dimension 44, where some topological manifolds have no triangulation.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

Show the path as a graph
Circle, concept, no evidence levelCircleconceptClosed ball, concept, no evidence levelClosed ballconceptConnected space, concept, no evidence levelConnected spaceconceptEuclidean space, object, no evidence levelEuclidean spaceobjectHomeomorphism, concept, no evidence levelHomeomorphismconceptTopological embedding, concept, no evidence levelTopological embeddingconceptTopological manifold, concept, no evidence levelTopological manifoldconceptTriangulation, concept, no evidence levelTriangulationconceptn-sphere, object, no evidence leveln-sphereobjectJordan curve theorem, statement, no evidence levelJordan curve theoremstatementSchoenflies theorem, statement, no evidence levelSchoenflies theoremstatementRadó's theorem on triangulation of surfaces, statement, no evidence levelRadó's theorem on tri…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Circle concept
  2. Closed ball concept
  3. Connected space concept
  4. Euclidean space object
  5. Homeomorphism concept
  6. Topological embedding concept
  7. Topological manifold concept
  8. Triangulation concept
  9. n-sphere object
  10. Jordan curve theorem statement
  11. Schoenflies theorem statement
  12. Radó's theorem on triangulation of surfaces statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Triangulation_(topology)
  2. Web https://en.wikipedia.org/wiki/Rad%C3%B3%27s_theorem_(Riemann_surfaces)
About this page
TagC9X05EDS5F
URIm2://C9X05EDS5F
Kindstatement, theorem
WritingW0 stub
MSC57Q15, 57K20
StatusThis copy of the site has no build data. Run m2 status C9X05EDS5F.
AboutTriangulation
UsesTopological manifold, Triangulation, Schoenflies theorem
Changed2 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by