Math 2.0

General topology Proof

Proof of the description of the open sets of a metric space

Builds on Metric spaceOpen setMetric topologyOpen ball

Proof

We unfold the definitions.

  1. Let (X,d)(X, d) be a metric space and let U⊆XU \subseteq X.
  2. By the definition of an open set of a metric space, UU is open in XX if and only if U∈τdU \in \tau_d. Here τd\tau_d is the metric topology.
  3. By the definition of τd\tau_d, we have U∈τdU \in \tau_d if and only if each x∈Ux \in U has a real ε>0\varepsilon > 0 with B(x,ε)⊆UB(x, \varepsilon) \subseteq U.
  4. Here B(x,ε)B(x, \varepsilon) is the open ball with centre xx and radius ε\varepsilon.
  5. Steps 2 and 3 together give: UU is open if and only if each x∈Ux \in U has an ε>0\varepsilon > 0 with B(x,ε)⊆UB(x, \varepsilon) \subseteq U.

Idea

The open sets of a metric space are by definition the members of the metric topology. That topology is defined by the ball condition, so the statement only puts the two definitions side by side.

Reading path

To understand this page from the ground up, read these 27 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 levelFunctiondefinitionMetric space, definition, no evidence levelMetric spacedefinitionOpen ball, definition, no evidence levelOpen balldefinitionMetric 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…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. Metric topology definition
  26. Open set definition
  27. Proof of the description of the open sets of a metric space proof

Where this leads

812 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Open_set
About this page
TagHFREZDYZH4
URIm2://HFREZDYZH4
Kindproof
WritingW0 stub
MSC54E35
ProvesOpen sets of a metric space
UsesMetric space, Open set, Metric topology, Open ball
Changed4 November 2026

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