Math 2.0

General topology Definition

One-point compactification

Builds on Topological spaceCompact setClosed setLocally compact spaceHausdorff space

Definition

Let (X,τ)(X, \tau) be a topological space and let ∞\infty be a point not in XX. Let X∗=X∪{∞}X^* = X \cup \{\infty\}. The one-point compactification, or Alexandroff extension, of XX is the set X∗X^* with the following topology. The open sets are the sets U∈τU \in \tau and the sets X∗∖CX^* \setminus C, where C⊆XC \subseteq X is closed and compact in XX.

Remarks

The space X∗X^* is compact, and XX is an open subspace of X∗X^*. The space X∗X^* is Hausdorff if and only if XX is locally compact and Hausdorff. The one-point compactification of ℝn\mathbb{R}^n is homeomorphic to the sphere SnS^n, by stereographic projection. For n=1n = 1, it is the circle.

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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 relationdefinitionUniversal set, definition, no evidence levelUniversal setdefinitionTopological space, definition, no evidence levelTopological spacedefinitionComplement, definition, no evidence levelComplementdefinitionFunction, definition, no evidence levelFunctiondefinitionSubspace topology, definition, no evidence levelSubspace topologydefinitionMetric space, definition, no evidence levelMetric spacedefinitionOpen ball, definition, no evidence levelOpen balldefinitionMetric topology, definition, no evidence levelMetric topologydefinitionOpen set, definition, no evidence levelOpen setdefinitionClosed set, definition, no evidence levelClosed setdefinitionHausdorff space, definition, no evidence levelHausdorff spacedefinitionOpen cover, definition, no evidence levelOpen coverdefinitionOpen sets characterised by neighbourhoods, statement, no evidence levelOpen sets characteris…statementCompact space, definition, no evidence levelCompact spacedefinitionCompact set, definition, no evidence levelCompact setdefinitionLocally compact space, definition, no evidence levelLocally compact spacedefinitionOne-point compactification, definition, no evidence levelOne-point compactific…definition
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. Universal set definition
  22. Topological space definition
  23. Complement definition
  24. Function definition
  25. Subspace topology definition
  26. Metric space definition
  27. Open ball definition
  28. Metric topology definition
  29. Open set definition
  30. Closed set definition
  31. Hausdorff space definition
  32. Open cover definition
  33. Open sets characterised by neighbourhoods statement
  34. Compact space definition
  35. Compact set definition
  36. Locally compact space definition
  37. One-point compactification definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Alexandroff_extension, The Alexandroff extension
About this page
Tag0MM45BT6HC
URIm2://0MM45BT6HC
Kinddefinition
WritingW1 stated
MSC54D35
DefinesOne-point compactification
UsesTopological space, Compact set, Closed set, Locally compact space, Hausdorff space

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