Math 2.0

Differential geometry Definition

Busemann G-space

Builds on Metric spaceProper metric spaceOpen ball

Definition

Let (X,d)(X, d) be a metric space. For points x,y,z∈Xx, y, z \in X, say that yy lies between xx and zz when y≠xy \ne x, y≠zy \ne z and d(x,y)+d(y,z)=d(x,z)d(x, y) + d(y, z) = d(x, z). The space XX is a Busemann G-space when it satisfies these four conditions.

  1. Finite compactness: every bounded closed subset of XX is compact.
  2. Menger convexity: for all distinct x,z∈Xx, z \in X there is a point between xx and zz.
  3. Local prolongability: each p∈Xp \in X has a radius rp>0r_p > 0 with this property. For distinct x,yx, y in the open ball B(p,rp)B(p, r_p), some z∈Xz \in X has yy between xx and zz.
  4. Uniqueness of prolongation: if yy lies between xx and z1z_1 and between xx and z2z_2, and d(y,z1)=d(y,z2)d(y, z_1) = d(y, z_2), then z1=z2z_1 = z_2.

Remarks

Condition 1 says that XX is a proper metric space. Example: the Euclidean plane ℝ2\R^2. It is proper, the midpoint lies between any two distinct points, and a segment from xx through yy extends along its line. The extension of length tt beyond yy is unique, since equality in the triangle inequality forces zz onto the ray from xx through yy.

Non-example: ℝ2\R^2 with the taxicab distance d1d_1 fails condition 4. Take x=(0,0)x = (0, 0), y=(1,0)y = (1, 0), z1=(2,0)z_1 = (2, 0) and z2=(32,12)z_2 = (\frac{3}{2}, \frac{1}{2}). Then d1(y,z1)=d1(y,z2)=1d_1(y, z_1) = d_1(y, z_2) = 1 and d1(x,z1)=d1(x,z2)=2d_1(x, z_1) = d_1(x, z_2) = 2, so yy lies between xx and both points, but z1≠z2z_1 \ne z_2.

Reading path

To understand this page from the ground up, read these 33 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionEmpty set, object, no evidence levelEmpty setobjectFamily of sets, definition, no evidence levelFamily of setsdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionIntersection, definition, no evidence levelIntersectiondefinitionUnion, definition, no evidence levelUniondefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionTopological space, definition, no evidence levelTopological spacedefinitionFunction, definition, no evidence levelFunctiondefinitionSubspace topology, definition, no evidence levelSubspace topologydefinitionMetric space, definition, no evidence levelMetric spacedefinitionClosed ball, definition, no evidence levelClosed balldefinitionOpen ball, definition, no evidence levelOpen balldefinitionMetric topology, definition, no evidence levelMetric topologydefinitionOpen set, definition, no evidence levelOpen setdefinitionOpen cover, definition, no evidence levelOpen coverdefinitionCompact space, definition, no evidence levelCompact spacedefinitionCompact set, definition, no evidence levelCompact setdefinitionProper metric space, definition, no evidence levelProper metric spacedefinitionBusemann G-space, definition, no evidence levelBusemann G-spacedefinition
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. Element definition
  2. Proposition concept
  3. Real numbers object
  4. Set definition
  5. Conjunction definition
  6. Disjunction definition
  7. Implication definition
  8. Predicate definition
  9. Empty set object
  10. Family of sets definition
  11. Ordered pair definition
  12. Intersection definition
  13. Union definition
  14. Existential quantifier definition
  15. Universal quantifier definition
  16. Cartesian product definition
  17. Subset definition
  18. Union of a family of sets definition
  19. Uniqueness quantifier definition
  20. Binary relation definition
  21. Topological space definition
  22. Function definition
  23. Subspace topology definition
  24. Metric space definition
  25. Closed ball definition
  26. Open ball definition
  27. Metric topology definition
  28. Open set definition
  29. Open cover definition
  30. Compact space definition
  31. Compact set definition
  32. Proper metric space definition
  33. Busemann G-space definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Busemann_G-space
About this page
Tag1EHBR0VMH3
URIm2://1EHBR0VMH3
Kinddefinition
WritingW0 stub
MSC53C70, 51F99
DefinesBusemann G-space
UsesMetric space, Proper metric space, Open ball

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