Math 2.0

General topology Proposition

Open balls form a basis of the metric topology

Builds on Basis of a topologyMetric topologyOpen balls are openOpen ball

Statement

Let (X,d)(X, d) be a metric space. Then the set of all open balls B(x,r)B(x, r), with x∈Xx \in X and real r>0r > 0, is a basis of the metric topology of XX. Also, the balls B(x,1/n)B(x, 1/n) with x∈Xx \in X and integer n≥1n \ge 1 form a basis.

Remarks

Proof: each ball is open, by open balls are open. If UU is open and x∈Ux \in U, then some ball B(x,r)B(x, r) lies in UU, and B(x,1/n)⊆B(x,r)B(x, 1/n) \subseteq B(x, r) for n>1/rn > 1/r. For each point xx, the balls B(x,1/n)B(x, 1/n) form a countable neighbourhood basis at xx. Thus each metric space is first-countable. In ℝ\mathbb{R}, the open balls are the bounded open intervals.

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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 spacedefinitionBasis of a topology, definition, no evidence levelBasis of a topologydefinitionFunction, definition, no evidence levelFunctiondefinitionMetric space, definition, no evidence levelMetric spacedefinitionOpen ball, definition, no evidence levelOpen balldefinitionTriangle inequality for a metric, statement, no evidence levelTriangle inequality f…statementMetric topology, definition, no evidence levelMetric topologydefinitionOpen set, definition, no evidence levelOpen setdefinitionOpen sets of a metric space, statement, no evidence levelOpen sets of a metric…statementOpen balls are open, statement, no evidence levelOpen balls are openstatementOpen balls form a basis of the metric topology, statement, no evidence levelOpen balls form a bas…statement
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.
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  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. Basis of a topology definition
  23. Function definition
  24. Metric space definition
  25. Open ball definition
  26. Triangle inequality for a metric statement
  27. Metric topology definition
  28. Open set definition
  29. Open sets of a metric space statement
  30. Open balls are open statement
  31. Open balls form a basis of the metric topology statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Base_(topology), Examples
About this page
Tag02EEX96DFS
URIm2://02EEX96DFS
Kindstatement, proposition
WritingW0 stub
MSC54E35
StatusThis copy of the site has no build data. Run m2 status 02EEX96DFS.
UsesBasis of a topology, Metric topology, Open balls are open, Open ball

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