Math 2.0

General topology Definition

Closed map

Builds on Topological spaceClosed setImage of a set

Definition

Let XX and YY be topological spaces. A function f ⁣:X→Yf \colon X \to Y is a closed map when for each closed set FF of XX, the image f(F)f(F) is closed in YY.

Remarks

A continuous map from a compact space to a Hausdorff space is closed. This gives continuous bijections from compact to Hausdorff spaces are homeomorphisms. The projection ℝ2→ℝ\mathbb{R}^2 \to \mathbb{R}, (x,y)↦x(x, y) \mapsto x, is open but not closed: it maps the closed set {(x,y):xy=1}\{(x, y) : xy = 1\} to ℝ∖{0}\mathbb{R} \setminus \{0\}. A map can be closed but not open. A constant map ℝ→ℝ\mathbb{R} \to \mathbb{R} is closed, but the image of ℝ\mathbb{R} is a point, which is not open.

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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 relationdefinitionUniversal set, definition, no evidence levelUniversal setdefinitionTopological space, definition, no evidence levelTopological spacedefinitionComplement, definition, no evidence levelComplementdefinitionFunction, definition, no evidence levelFunctiondefinitionImage of a set, definition, no evidence levelImage 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 setdefinitionClosed set, definition, no evidence levelClosed setdefinitionClosed map, definition, no evidence levelClosed mapdefinition
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. Universal set definition
  22. Topological space definition
  23. Complement definition
  24. Function definition
  25. Image of a set definition
  26. Metric space definition
  27. Open ball definition
  28. Metric topology definition
  29. Open set definition
  30. Closed set definition
  31. Closed map definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Open_and_closed_maps, Definitions and characterizations
About this page
Tag0NYZ2M45G9
URIm2://0NYZ2M45G9
Kinddefinition
WritingW0 stub
MSC54C10
DefinesClosed map
UsesTopological space, Closed set, Image of a set

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