Math 2.0

General topology Definition

Box topology

Builds on Topological spaceBasis of a topologyCartesian product

Definition

Let (Xi)i∈I(X_i)_{i \in I} be a family of topological spaces and let X=∏i∈IXiX = \prod_{i \in I} X_i be the Cartesian product. The box topology on XX is the topology with basis the set of all products ∏i∈IUi\prod_{i \in I} U_i, where each UiU_i is open in XiX_i.

Remarks

For finitely many factors, the box topology is the product topology. For infinite products, the box topology is finer than the product topology, and often strictly finer. On ℝℕ\mathbb{R}^{\mathbb{N}} with the box topology, the map t↦(t,t,t,…)t \mapsto (t, t, t, \ldots) from ℝ\mathbb{R} is not continuous, although each coordinate is continuous. The preimage of ∏n≥1(−1/n,1/n)\prod_{n \ge 1} (-1/n, 1/n) is {0}\{0\}, which is not open.

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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 levelSubsetdefinitionUnion 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 topologydefinitionBox topology, definition, no evidence levelBox topologydefinition
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. Union of a family of sets definition
  18. Topological space definition
  19. Basis of a topology definition
  20. Box topology definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Box_topology, Definition
About this page
Tag0CTM1EHG9P
URIm2://0CTM1EHG9P
Kinddefinition
WritingW1 stated
MSC54B10
DefinesBox topology
UsesTopological space, Basis of a topology, Cartesian product

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