Math 2.0

General topology Definition

Product topology

Builds on Topological spaceBasis of a topologyCartesian product

Definition

Let (X,τX)(X, \tau_X) and (Y,τY)(Y, \tau_Y) be topological spaces. The product topology on the Cartesian product X×YX \times Y is the topology with basis {U×V:U∈τX, V∈τY}.\{ U \times V : U \in \tau_X, \ V \in \tau_Y \}.

Remarks

This family is a basis of a topology, because (U1×V1)∩(U2×V2)=(U1∩U2)×(V1∩V2)(U_1 \times V_1) \cap (U_2 \times V_2) = (U_1 \cap U_2) \times (V_1 \cap V_2). A set W⊆X×YW \subseteq X \times Y is open if and only if for each (x,y)∈W(x, y) \in W there are open U∋xU \ni x and V∋yV \ni y with U×V⊆WU \times V \subseteq W. For metric spaces, it is the metric topology of the product metric. The product topology on ℝ×ℝ\mathbb{R} \times \mathbb{R} is the usual topology of ℝ2\mathbb{R}^2. The projections are continuous. For products of any family of spaces, see the product topology on an arbitrary product.

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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 topologydefinitionProduct topology, definition, no evidence levelProduct topologydefinition
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  1. Element definition
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  14. Universal quantifier definition
  15. Cartesian product definition
  16. Subset definition
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  18. Topological space definition
  19. Basis of a topology definition
  20. Product topology definition

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27 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Product_topology, Definition
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Kinddefinition
WritingW1 stated
MSC54B10
DefinesProduct topology
UsesTopological space, Basis of a topology, Cartesian product

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