connected space
A topological space is said to be connected if there is no pair of nonempty subsets such that both and are open in , and . If is not connected, i.e. if there are sets and with the above properties, then we say that is disconnected.
Every topological space can be viewed as a collection of subspaces each of which are connected. These subspaces are called the connected components of . Slightly more rigorously, we define an equivalence relation on points in by declaring that if there is a connected subset of such that and both lie in . Then a connected component of 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 |