Beweis der Invarianz der Dimensionenzahl¶
Brouwer, L. E. J. (1912). Beweis der Invarianz der Dimensionenzahl. Mathematische Annalen, 70(2), 161-165.
Cited by¶
3 citations across 3 artifacts.
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Primes¶
- Continuity
- Dimension
- generalized this to n-dimensional differentiable manifolds with variable curvature, laying the groundwork for general relativity and modern differential geometry; Brouwer's 1911 invariance-of-dimension theorem
This sourceEstablishes topological invariance of dimension under homeomorphism — ℝᵐ and ℝⁿ are not homeomorphic for m≠n.
- generalized this to n-dimensional differentiable manifolds with variable curvature, laying the groundwork for general relativity and modern differential geometry; Brouwer's 1911 invariance-of-dimension theorem
- Topology
- A topological obstruction to a desired transformation — a non-trivial topological invariant that any candidate transformation would have to map to itself, which it cannot — proves that the transformation does not exist, regardless of how much metric flexibility one has in constructing it.
This sourceThe 1911 paper establishes the topological invariance of dimension under homeomorphism (no continuous bijection exists between Euclidean spaces of different dimensions); the 1912 paper develops the degree of a map and proves the Brouwer fixed-point theorem (every continuous self-map of a closed disk has a fixed point), a paradigm topological-impossibility result.
- A topological obstruction to a desired transformation — a non-trivial topological invariant that any candidate transformation would have to map to itself, which it cannot — proves that the transformation does not exist, regardless of how much metric flexibility one has in constructing it.
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