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∩V=∅ and U∪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
∼ on points in X by declaring that x∼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 |