Math 2.0

General topology Proposition

Open sets characterised by neighbourhoods

Builds on Topological spaceOpen setUnion of a family of sets

Statement

Let (X,τ)(X, \tau) be a topological space and let x∈Xx \in X. A set N⊆XN \subseteq X is a neighbourhood of xx when there is an open set UU with x∈U⊆Nx \in U \subseteq N. Then a set A⊆XA \subseteq X is open if and only if AA is a neighbourhood of each of its points.

Remarks

Proof of the converse: for each x∈Ax \in A, choose an open UxU_x with x∈Ux⊆Ax \in U_x \subseteq A. Then A=⋃x∈AUxA = \bigcup_{x \in A} U_x, which is open as a union of open sets. A neighbourhood need not be open. For example, [−1,1][-1, 1] is a neighbourhood of 00 in ℝ\mathbb{R}. In a metric space, NN is a neighbourhood of xx if and only if NN contains a ball B(x,ε)B(x, \varepsilon) for some ε>0\varepsilon > 0.

Reading path

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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 setdefinitionOpen sets characterised by neighbourhoods, statement, no evidence levelOpen sets characteris…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. Function definition
  23. Metric space definition
  24. Open ball definition
  25. Metric topology definition
  26. Open set definition
  27. Open sets characterised by neighbourhoods statement

Where this leads

25 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Neighbourhood_(mathematics), Neighbourhood of a point
  2. Web https://en.wikipedia.org/wiki/Neighbourhood_(mathematics), Topology from neighbourhoods
About this page
Tag09CJ00H3NW
URIm2://09CJ00H3NW
Kindstatement, proposition
WritingW0 stub
MSC54A05
StatusThis copy of the site has no build data. Run m2 status 09CJ00H3NW.
UsesTopological space, Open set, Union of a family of sets

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