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connected space (Definition)

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.



"connected space" is owned by mathcam. [ full author list (2) | owner history (1) ]
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See Also: semilocally simply connected, extremally disconnected, example of a connected space that is not path-connected, locally connected, proof of generalized intermediate value theorem, a connected normal space with more than one point is uncountable, (path) connectness as a homotopy invariant

Also defines:  connected, connected components, disconnected, connectedness

Attachments:
connectedness is preserved under a continuous map (Theorem) by drini
products of connected spaces are connected (Theorem) by mps
cut-point (Definition) by mathcam
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Cross-references: relation, equivalence class, points, equivalence relation, subspaces, collection, properties, open, subsets, topological space
There are 97 references to this entry.

This is version 12 of connected space, born on 2001-11-17, modified 2006-08-10.
Object id is 941, canonical name is ConnectedSpace.
Accessed 14606 times total.

Classification:
AMS MSC: 54D05 (General topology :: Fairly general properties :: Connected and locally connected spaces )
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