connectedness is preserved under a continuous map

Theorem Suppose f:XY is a continuous mapMathworldPlanetmath between topological spacesMathworldPlanetmath X and Y. If X is a connected space, and f is surjectivePlanetmathPlanetmath, then Y is a connected space.

The inclusion mapMathworldPlanetmath for spaces X=(0,1) and Y=(0,1)(2,3) shows that we need to assume that the map is surjective. Othewise, we can only prove that f(X) is connected. See this page (

Proof. For a contradictionMathworldPlanetmathPlanetmath, suppose there are disjoint open sets A,B in Y such that Y=AB. By continuity and properties of the inverse image, f-1(A) and f-1(B) are open disjoint sets in X. Since f is surjective, Y=f(X)=AB, whence


contradicting the assumptionPlanetmathPlanetmath that X is connected.


Title connectedness is preserved under a continuous map
Canonical name ConnectednessIsPreservedUnderAContinuousMap
Date of creation 2013-03-22 13:55:59
Last modified on 2013-03-22 13:55:59
Owner drini (3)
Last modified by drini (3)
Numerical id 7
Author drini (3)
Entry type Theorem
Classification msc 54D05
Related topic CompactnessIsPreservedUnderAContinuousMap
Related topic ProofOfGeneralizedIntermediateValueTheorem