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 $X$, $U\cap V=\mathrm{\varnothing }$ 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