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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, 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
intervals are connected (Example) by joking
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Cross-references: relation, equivalence class, points, equivalence relation, subspaces, collection, properties, open, subsets, topological space
There are 123 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 19204 times total.

Classification:
AMS MSC54D05 (General topology :: Fairly general properties :: Connected and locally connected spaces )

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