products of connected spaces are connected

Theorem 1

[1, 2] Let (Xi)iI be a family of topological spacesMathworldPlanetmath. Then the product space


with the product topology is connected if and only if each space Xi is connected.

As is true of most results in topology involving productsPlanetmathPlanetmath, the forward implicationMathworldPlanetmath requires the axiom of choiceMathworldPlanetmath.


  • 1 S. Lang, AnalysisMathworldPlanetmath II, Addison-Wesley Publishing Company Inc., 1969.
  • 2 A. Mukherjea, K. Pothoven, Real and Functional AnalysisMathworldPlanetmath, Plenum Press, 1978.
Title products of connected spaces are connected
Canonical name ProductsOfConnectedSpacesAreConnected
Date of creation 2013-03-22 13:56:13
Last modified on 2013-03-22 13:56:13
Owner mps (409)
Last modified by mps (409)
Numerical id 6
Author mps (409)
Entry type Theorem
Classification msc 54D05