Math 2.0

Manifolds and cell complexes Definition

Manifold with boundary

Builds on Topological spaceHausdorff spaceSecond-countable spaceOpen setHomeomorphismEuclidean spaceTopological manifoldInvariance of domain

Definition

Let n≥1n \ge 1 be an integer and let ℍn={x∈ℝn:xn≥0}\mathbb{H}^n = \set{x \in \R^n : x_n \ge 0} be the closed upper half-space of Euclidean space, with the subspace topology. A topological space MM is a topological manifold with boundary of dimension nn if:

  1. MM is Hausdorff.
  2. MM is second-countable.
  3. Each point x∈Mx \in M has an open neighbourhood that is homeomorphic to an open subset of ℍn\mathbb{H}^n.

A point x∈Mx \in M is a boundary point if some homeomorphism φ\varphi from an open neighbourhood of xx onto an open subset of ℍn\mathbb{H}^n has φ(x)∈{xn=0}\varphi(x) \in \set{x_n = 0}. The set of boundary points is the boundary ∂M\partial M, and M∖∂MM \setminus \partial M is the interior of MM.

Remarks

The closed disk D2={x∈ℝ2:∣x∣≤1}D^2 = \set{x \in \R^2 : \abs{x} \le 1} is a manifold with boundary of dimension 22, with ∂D2=S1\partial D^2 = S^1. Near a point of the open disk it is an open subset of ℝ2\R^2, which is homeomorphic to an open subset of the interior of ℍ2\mathbb{H}^2. Near a point of S1S^1, polar coordinates (r,θ)↦(θ,1−r)(r, \theta) \mapsto (\theta, 1 - r) map a neighbourhood onto an open subset of ℍ2\mathbb{H}^2, sending S1S^1 into the line {x2=0}\set{x_2 = 0}.

The cross {(x,y)∈ℝ2:xy=0}\set{(x, y) \in \R^2 : xy = 0} satisfies clauses 1 and 2 but fails clause 3 at the origin, where four arcs meet.

By invariance of domain, no point is a boundary point for one chart and an interior point for another, so the boundary is well defined. The interior is a topological manifold of dimension nn, and ∂M\partial M is a topological manifold of dimension n−1n - 1 with empty boundary. A topological manifold is the same as a manifold with boundary whose boundary is empty. Some authors use {x1≥0}\set{x_1 \ge 0} or {x1≤0}\set{x_1 \le 0} as the model half-space; the notion does not change.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

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Continuous map, concept, no evidence levelContinuous mapconceptEuclidean space, object, no evidence levelEuclidean spaceobjectHausdorff space, concept, no evidence levelHausdorff spaceconceptHomeomorphism, concept, no evidence levelHomeomorphismconceptInjective function, concept, no evidence levelInjective functionconceptOpen set, concept, no evidence levelOpen setconceptSecond-countable space, concept, no evidence levelSecond-countable spaceconceptTopological manifold, concept, no evidence levelTopological manifoldconceptTopological space, concept, no evidence levelTopological spaceconceptInvariance of domain, statement, no evidence levelInvariance of domainstatementManifold with boundary, definition, no evidence levelManifold with boundarydefinition
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. Continuous map concept
  2. Euclidean space object
  3. Hausdorff space concept
  4. Homeomorphism concept
  5. Injective function concept
  6. Open set concept
  7. Second-countable space concept
  8. Topological manifold concept
  9. Topological space concept
  10. Invariance of domain statement
  11. Manifold with boundary definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Manifold, Manifold with boundary
About this page
TagYZ8AN4J2KD
URIm2://YZ8AN4J2KD
Kinddefinition
WritingW0 stub
MSC57N99
DefinesManifold with boundary
UsesTopological space, Hausdorff space, Second-countable space, Open set, Homeomorphism, Euclidean space, Topological manifold, Invariance of domain
Changed2 November 2026

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