Math 2.0

General topology Compactness Proposition

Continuous image of a compact set is compact

Builds on Continuous map of topological spacesCompactness via open covers

Statement

Let XX and YY be topological spaces, let f ⁣:X→Yf \colon X \to Y be a continuous map and let K⊆XK \subseteq X be compact. Then the image f(K)f(K) is compact in YY.

Remarks

Proof: let (Vi)(V_i) be open sets of YY that cover f(K)f(K). The sets f−1(Vi)f^{-1}(V_i) are open and cover KK. A finite subfamily covers KK, and the matching ViV_i cover f(K)f(K). A consequence is that a continuous real function on a compact set attains its bounds. Compactness is a topological property.

Reading path

To understand this page from the ground up, read these 34 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 topologydefinitionPreimage of a set, definition, no evidence levelPreimage of a setdefinitionMetric space, definition, no evidence levelMetric spacedefinitionOpen ball, definition, no evidence levelOpen balldefinitionMetric topology, definition, no evidence levelMetric topologydefinitionOpen set, definition, no evidence levelOpen setdefinitionContinuous map of topological spaces, definition, no evidence levelContinuous map of top…definitionOpen 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 …statementContinuous image of a compact set is compact, statement, no evidence levelContinuous image of a…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.
  • 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. Subspace topology definition
  24. Preimage of a set definition
  25. Metric space definition
  26. Open ball definition
  27. Metric topology definition
  28. Open set definition
  29. Continuous map of topological spaces definition
  30. Open cover definition
  31. Compact space definition
  32. Compact set definition
  33. Compactness via open covers statement
  34. Continuous image of a compact set is compact statement

Where this leads

28 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Compact_space, Functions and compact spaces
About this page
Tag0K9AAJ8G51
URIm2://0K9AAJ8G51
Kindstatement, proposition
WritingW0 stub
MSC54D30
StatusThis copy of the site has no build data. Run m2 status 0K9AAJ8G51.
UsesContinuous map of topological spaces, Compactness via open covers

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