Math 2.0

General topology Definition

T1 space

Builds on Topological spaceOpen setClosed set

Definition

A topological space XX is T1 when for all points x≠yx \neq y of XX there is an open set UU with x∈Ux \in U and y∉Uy \notin U.

Remarks

A space is T1 if and only if each set {x}\{x\} with x∈Xx \in X is closed. Each Hausdorff space is T1, and each T1 space is T0. An infinite set with the cofinite topology is T1 but not Hausdorff. In fact, the cofinite topology is the coarsest T1 topology on a set.

Reading path

To understand this page from the ground up, read these 30 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 levelFunctiondefinitionMetric 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 setdefinitionT1 space, definition, no evidence levelT1 spacedefinition
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. Metric space definition
  26. Open ball definition
  27. Metric topology definition
  28. Open set definition
  29. Closed set definition
  30. T1 space definition

Further reading

  1. Web https://en.wikipedia.org/wiki/T1_space, Definitions
About this page
Tag05GQJXBP3F
URIm2://05GQJXBP3F
Kinddefinition
WritingW1 stated
MSC54D10
DefinesT1 space
UsesTopological space, Open set, Closed set

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

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

What links here

No page links to this page yet.