Math 2.0

General topology Definition

Open cover

Builds on Open setUnion of a family of sets

Definition

Let XX be a topological space and let A⊆XA \subseteq X. An open cover of AA is a family (Ui)i∈I(U_i)_{i \in I} of open subsets of XX with A⊆⋃i∈IUiA \subseteq \bigcup_{i \in I} U_i. A subcover is a subfamily (Uj)j∈J(U_j)_{j \in J}, with J⊆IJ \subseteq I, that is still an open cover of AA. A subcover is finite when JJ is a finite set.

Remarks

The intervals (−n,n)(-n, n) for n≥1n \ge 1 form an open cover of ℝ\mathbb{R}. It has no finite subcover. The intervals (1/n,1)(1/n, 1) for n≥2n \ge 2 form an open cover of (0,1)(0, 1) with no finite subcover. A set KK is compact when each open cover of KK has a finite subcover.

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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 cover, definition, no evidence levelOpen coverdefinition
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. Open cover definition

Where this leads

90 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Cover_(topology), Definition
  2. Web https://en.wikipedia.org/wiki/Cover_(topology), Subcover
About this page
Tag0NVPNC12JY
URIm2://0NVPNC12JY
Kinddefinition
WritingW1 stated
MSC54D30
DefinesOpen cover
UsesOpen set, Union of a family of sets

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