PlanetMath: connected component

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connected component

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Two points $x,y$ in a topological space $X$ are said to be in the same connected component if there exists a subspace of $X$ containing $x$ and $y$ which is connected. This relation is an equivalence relation, and the equivalence classes of $X$ under this relation are called the connected components of $X$

"connected component" is owned by djao. [ full author list (2) ]

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Attachments:: homeomorphisms preserve connected components (Derivation) by joking

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Cross-references: equivalence classes, equivalence relation, relation, connected, subspace, connected component, topological space, points
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This is version 2 of connected component, born on 2001-11-17, modified 2002-08-22.
Object id is 948, canonical name is ConnectedComponent.
Accessed 3062 times total.

Classification:

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

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