Math 2.0

General topology Proposition

Compactness via open covers

Builds on Compact setOpen coverSubspace topology

Statement

Let XX be a topological space and let K⊆XK \subseteq X. Then KK is compact if and only if each family (Ui)i∈I(U_i)_{i \in I} of open subsets of XX with K⊆⋃i∈IUiK \subseteq \bigcup_{i \in I} U_i has a finite subfamily Ui1,…,UinU_{i_1}, \ldots, U_{i_n} with K⊆Ui1∪⋯∪UinK \subseteq U_{i_1} \cup \cdots \cup U_{i_n}.

Remarks

Proof: the open sets of the subspace topology on KK are the sets U∩KU \cap K with UU open in XX. So covers of KK by open sets of KK and covers by open sets of XX match each other. For K=XK = X, the statement is the definition of a compact space. Thus compactness of a subset does not depend on the space around it. If K⊆Y⊆XK \subseteq Y \subseteq X, then KK is compact in YY if and only if it is compact in XX.

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To understand this page from the ground up, read these 31 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 levelFunctiondefinitionSubspace topology, definition, no evidence levelSubspace topologydefinitionMetric 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 coverdefinitionCompact space, definition, no evidence levelCompact spacedefinitionCompact set, definition, no evidence levelCompact setdefinitionCompactness via open covers, statement, no evidence levelCompactness via open …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. Subspace topology definition
  24. Metric space definition
  25. Open ball definition
  26. Metric topology definition
  27. Open set definition
  28. Open cover definition
  29. Compact space definition
  30. Compact set definition
  31. Compactness via open covers statement

Where this leads

36 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Compact_space, Compactness of subsets
About this page
Tag0P6PMJB4MA
URIm2://0P6PMJB4MA
Kindstatement, proposition
WritingW0 stub
MSC54D30
StatusThis copy of the site has no build data. Run m2 status 0P6PMJB4MA.
UsesCompact set, Open cover, Subspace topology

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