Math 2.0

General topology Definition

Open set

Builds on Topological spaceMetric topology

Definition

Let (X,τ)(X, \tau) be a topological space. A subset UU of XX is open in XX when U∈τU \in \tau. A subset of a metric space is open when it is open for the metric topology.

Remarks

In a metric space (X,d)(X, d), a set UU is open if and only if for each x∈Ux \in U there is ε>0\varepsilon > 0 with B(x,ε)⊆UB(x, \varepsilon) \subseteq U. This is the statement open sets of a metric space. In ℝ\mathbb{R}, each open interval (a,b)(a, b) is open, but [0,1)[0, 1) is not open: no ball about 00 lies in [0,1)[0, 1). The empty set and XX are always open.

Reading path

To understand this page from the ground up, read these 26 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 setdefinition
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

Where this leads

263 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Open_set, Definitions
About this page
Tag01ZQQDCERP
URIm2://01ZQQDCERP
Kinddefinition
WritingW1 stated
MSC54A05
DefinesOpen set
UsesTopological space, Metric topology

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