Math 2.0

General topology Compactness Proposition

Compact subsets of a Hausdorff space are closed

Builds on Compactness via open coversHausdorff spaceClosed set

Statement

Let XX be a Hausdorff space and let KK be a compact subset of XX. Then KK is closed in XX.

Remarks

Proof: fix y∉Ky \notin K. For each x∈Kx \in K, choose disjoint open sets Ux∋xU_x \ni x and Vx∋yV_x \ni y. Finitely many Ux1,…,UxnU_{x_1}, \ldots, U_{x_n} cover KK. Then Vx1∩⋯∩VxnV_{x_1} \cap \cdots \cap V_{x_n} is an open set that contains yy and misses KK. The Hausdorff condition is needed. With the indiscrete topology on a set with two points, each one-point set is compact but not closed. Each metric space is Hausdorff, so compact subsets of metric spaces are closed.

Reading path

To understand this page from the ground up, read these 36 pages in order. The foundations come first.

Show the path as a graph
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 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 setdefinitionClosed set, definition, no evidence levelClosed setdefinitionHausdorff space, definition, no evidence levelHausdorff spacedefinitionOpen 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 …statementCompact subsets of a Hausdorff space are closed, statement, no evidence levelCompact subsets 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. Universal set definition
  22. Topological space definition
  23. Complement definition
  24. Function definition
  25. Subspace topology definition
  26. Metric space definition
  27. Open ball definition
  28. Metric topology definition
  29. Open set definition
  30. Closed set definition
  31. Hausdorff space definition
  32. Open cover definition
  33. Compact space definition
  34. Compact set definition
  35. Compactness via open covers statement
  36. Compact subsets of a Hausdorff space are closed statement

Where this leads

22 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Compact_space, Properties of compact spaces
About this page
Tag0MYFJVMN7M
URIm2://0MYFJVMN7M
Kindstatement, proposition
WritingW0 stub
MSC54D30
StatusThis copy of the site has no build data. Run m2 status 0MYFJVMN7M.
UsesCompactness via open covers, Hausdorff space, Closed set

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by