Math 2.0

General topology Definition

Product topology on an arbitrary product

Builds on Topological spaceSubbasisCartesian product

Definition

Let (Xi)i∈I(X_i)_{i \in I} be a family of topological spaces, indexed by any set II, and let X=∏i∈IXiX = \prod_{i \in I} X_i be the Cartesian product, with projections πi ⁣:X→Xi\pi_i \colon X \to X_i. The product topology on XX is the topology generated by the subbasis {πi−1(U):i∈I, U open in Xi}.\{ \pi_i^{-1}(U) : i \in I, \ U \text{ open in } X_i \}.

Remarks

A basis consists of the sets ∏i∈IUi\prod_{i \in I} U_i, where each UiU_i is open in XiX_i and Ui=XiU_i = X_i for all but finitely many ii. For two factors, this is the product topology of X1×X2X_1 \times X_2. It is the coarsest topology on XX for which each projection πi\pi_i is continuous. A sequence converges in XX if and only if each of its coordinates converges. The box topology is finer and lacks this property.

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To understand this page from the ground up, read these 22 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptSet, 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 levelSubsetdefinitionIntersection of a family of sets, definition, no evidence levelIntersection of a fam…definitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionTopological space, definition, no evidence levelTopological spacedefinitionBasis of a topology, definition, no evidence levelBasis of a topologydefinitionSubbasis, definition, no evidence levelSubbasisdefinitionProduct topology on an arbitrary product, definition, no evidence levelProduct topology on a…definition
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. Set definition
  4. Conjunction definition
  5. Disjunction definition
  6. Implication definition
  7. Predicate definition
  8. Empty set object
  9. Family of sets definition
  10. Ordered pair definition
  11. Intersection definition
  12. Union definition
  13. Existential quantifier definition
  14. Universal quantifier definition
  15. Cartesian product definition
  16. Subset definition
  17. Intersection of a family of sets definition
  18. Union of a family of sets definition
  19. Topological space definition
  20. Basis of a topology definition
  21. Subbasis definition
  22. Product topology on an arbitrary product definition

Where this leads

7 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Product_topology, Definition
About this page
Tag04S9HX3GAP
URIm2://04S9HX3GAP
Aliasesproduct-of-topological-spaces
Kinddefinition
WritingW1 stated
MSC54B10
DefinesProduct topology
UsesTopological space, Subbasis, Cartesian product

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