connected space

A topological spaceMathworldPlanetmath X is said to be connectedPlanetmathPlanetmathPlanetmath if there is no pair of nonempty subsets U,V such that both U and V are open in X, UV= and UV=X. If X is not connected, i.e. if there are sets U and V with the above properties, then we say that X is disconnected.

Every topological space X can be viewed as a collectionMathworldPlanetmath of subspacesMathworldPlanetmathPlanetmath each of which are connected. These subspaces are called the connected componentsMathworldPlanetmathPlanetmath of X. Slightly more rigorously, we define an equivalence relationMathworldPlanetmath on points in X by declaring that xy if there is a connected subset Y of X such that x and y both lie in Y. Then a connected component of X is defined to be an equivalence classMathworldPlanetmath under this relationMathworldPlanetmath.

Title connected space
Canonical name ConnectedSpace
Date of creation 2013-03-22 12:00:11
Last modified on 2013-03-22 12:00:11
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 16
Author mathcam (2727)
Entry type Definition
Classification msc 54D05
Related topic SemilocallySimplyConnected
Related topic ExtremallyDisconnected
Related topic ExampleOfAConnectedSpaceWhichIsNotPathConnected
Related topic LocallyConnected
Related topic ProofOfGeneralizedIntermediateValueTheorem
Related topic AConnectedNormalSpaceWithMoreThanOnePointIsUncountable2
Related topic AConnectedNormalSpaceWithMoreThanOn
Defines connected
Defines connected components
Defines disconnected
Defines connectedness