Math 2.0

Manifolds and cell complexes Definition

Topological manifold

Builds on Topological spaceHausdorff spaceSecond-countable spaceOpen setHomeomorphismEuclidean space

Definition

Let n≥0n \ge 0 be an integer and let MM be a topological space. MM is a topological manifold of dimension nn, or a topological nn-manifold, if:

  1. MM is Hausdorff.
  2. MM is second-countable.
  3. MM is locally Euclidean of dimension nn: each point x∈Mx \in M has an open neighbourhood UU homeomorphic to an open subset of ℝn\R^n, the Euclidean space.

Remarks

The unit circle S1={(x,y)∈ℝ2:x2+y2=1}S^1 = \set{(x, y) \in \R^2 : x^2 + y^2 = 1} is a 11-manifold. It is Hausdorff and second-countable as a subspace of ℝ2\R^2. The map (x,y)↦x(x, y) \mapsto x sends the open upper half {y>0}\set{y > 0} of S1S^1 homeomorphically onto the interval (−1,1)(-1, 1). The three other open half circles are treated in the same way, and together the four cover S1S^1.

The line with two origins is the quotient of ℝ×{0,1}\R \times \set{0, 1} obtained by identifying (t,0)(t, 0) with (t,1)(t, 1) for all t≠0t \ne 0. It satisfies clauses 2 and 3, but it fails clause 1: the two origins have no disjoint neighbourhoods.

A nonempty manifold has only one dimension, by invariance of domain: a nonempty open subset of ℝm\R^m is not homeomorphic to an open subset of ℝn\R^n when m≠nm \ne n. A pair (U,φ)(U, \varphi) with φ\varphi a homeomorphism from UU onto an open subset of ℝn\R^n is called a chart. A compact manifold without boundary is called a closed manifold.

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Euclidean space, object, no evidence levelEuclidean spaceobjectHausdorff space, concept, no evidence levelHausdorff spaceconceptHomeomorphism, concept, no evidence levelHomeomorphismconceptOpen set, concept, no evidence levelOpen setconceptSecond-countable space, concept, no evidence levelSecond-countable spaceconceptTopological space, concept, no evidence levelTopological spaceconceptTopological manifold, definition, no evidence levelTopological manifolddefinition
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  1. Euclidean space object
  2. Hausdorff space concept
  3. Homeomorphism concept
  4. Open set concept
  5. Second-countable space concept
  6. Topological space concept
  7. Topological manifold definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Topological_manifold, Formal definition
About this page
TagZ9DYVG5RA3
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Kinddefinition
WritingW1 stated
MSC57N99
DefinesTopological manifold
UsesTopological space, Hausdorff space, Second-countable space, Open set, Homeomorphism, Euclidean space
Changed2 November 2026

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