product map

Notation: If {Xi}iI is a collectionMathworldPlanetmath of sets (indexed by I) then iIXi denotes the generalized Cartesian product of {Xi}ıI.

Let {Ai}iI and {Bi}iI be collections of sets indexed by the same set I and fi:AiBi a collection of functions.

The product map is the function


0.1 Properties:

  • If fi:AiBi and gi:BiCi are collections of functions then

  • iIfi is injectivePlanetmathPlanetmath if and only if each fi is injective.

  • iIfi is surjectivePlanetmathPlanetmath if and only if each fi is surjective.

  • Suppose {Ai}iI and {Bi}iI are topological spacesMathworldPlanetmath. Then iIfi is continuousPlanetmathPlanetmath ( (in the product topology) if and only if each fi is continuous.

  • Suppose {Ai}iI and {Bi}iI are groups, or rings or algebras. Then iIfi is a group (ring or ) homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath if and only if each fi is a group (ring or ) homomorphism.

Title product map
Canonical name ProductMap
Date of creation 2013-03-22 17:48:28
Last modified on 2013-03-22 17:48:28
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 6
Author asteroid (17536)
Entry type Definition
Classification msc 03E20