# connected space

A topological space $X$ is said to be connected if there is no pair of nonempty subsets $U,V$ such that both $U$ and $V$ are open in $X$, $U\cap V=\emptyset$ and $U\cup V=X$. 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 collection of subspaces each of which are connected. These subspaces are called the connected components of $X$. Slightly more rigorously, we define an equivalence relation $\sim$ on points in $X$ by declaring that $x\sim y$ 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 class under this relation.

 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