Math 2.0

General topology Proof

Proof that open balls are open

Builds on Metric spaceOpen ballTriangle inequality for a metricOpen sets of a metric space

Proof

  1. Let (X,d)(X, d) be a metric space, let x∈Xx \in X and let r>0r > 0. Let B(x,r)={z∈X:d(x,z)<r}B(x, r) = \set{z \in X : d(x, z) < r} be the open ball.
  2. Let y∈B(x,r)y \in B(x, r). Then d(x,y)<rd(x, y) < r.
  3. Put ε=r−d(x,y)\varepsilon = r - d(x, y). By step 2, ε>0\varepsilon > 0.
  4. Let z∈B(y,ε)z \in B(y, \varepsilon), so d(y,z)<εd(y, z) < \varepsilon.
  5. By the triangle inequality, d(x,z)≤d(x,y)+d(y,z)<d(x,y)+ε=rd(x, z) \le d(x, y) + d(y, z) < d(x, y) + \varepsilon = r.
  6. So z∈B(x,r)z \in B(x, r). As zz was arbitrary, B(y,ε)⊆B(x,r)B(y, \varepsilon) \subseteq B(x, r).
  7. Each point yy of B(x,r)B(x, r) thus has a ball about it inside B(x,r)B(x, r).
  8. By the description of the open sets of a metric space, B(x,r)B(x, r) is open in XX.

Idea

A point yy inside the ball is at distance d(x,y)d(x, y) from the centre. So it still has room r−d(x,y)r - d(x, y) before it reaches the edge. The triangle inequality shows that the ball of that radius about yy stays inside.

Reading path

To understand this page from the ground up, read these 30 pages in order. The foundations come first. The path takes the proof Proof of the description of the open sets of a metric space.

Show the path as a graph
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 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 setdefinitionProof of the description of the open sets of a metric space, proof, no evidence levelProof of the descript…proofOpen sets of a metric space, statement, no evidence levelOpen sets of a metric…statementProof that open balls are open, proof, no evidence levelProof that open balls…proof
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. Metric space definition
  24. Open ball definition
  25. Triangle inequality for a metric statement
  26. Metric topology definition
  27. Open set definition
  28. Proof of the description of the open sets of a metric space proof
  29. Open sets of a metric space statement
  30. Proof that open balls are open proof

Where this leads

770 pages build on this one.

Further reading

  1. Web https://proofwiki.org/wiki/Open_Ball_of_Metric_Space_is_Open_Set
About this page
TagVPHQ121S8A
URIm2://VPHQ121S8A
Kindproof
WritingW0 stub
MSC54E35
ProvesOpen balls are open
UsesMetric space, Open ball, Triangle inequality for a metric, Open sets of a metric space
Changed4 November 2026

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