Connected component: Difference between revisions
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For a [[topological space]] <math>X</math>, consider the following relation: <math>a \sim b</math> if there exists a subset of <math>X</math> containing both <math>a</math> and <math>b</math> that is a [[connected space]] under the [[subspace topology]]. Then, it turns out that <math>\! \sim</math> is an equivalence relation on <math>X</math>. The equivalence classes under <math>\! \sim </math> are termed the connected components of <math>X</math>. | For a [[topological space]] <math>X</math>, consider the following relation: <math>a \sim b</math> if there exists a subset of <math>X</math> containing both <math>a</math> and <math>b</math> that is a [[connected space]] under the [[subspace topology]]. Then, it turns out that <math>\! \sim</math> is an equivalence relation on <math>X</math>. The equivalence classes under <math>\! \sim </math> are termed the connected components of <math>X</math>. | ||
The relation <matH>\! \sim</math> is termed the relation of being in the same connected component. | |||
===Equivalence of definitions=== | |||
{{further|[[equivalence of definitions of connected component]]}} | |||
==Facts== | |||
* The connected components are pairwise disjoint subsets. This follows from the equivalence relation version of the definition. | |||
* [[Connected components are closed]]. This follows from the fact that the closure of a connected subset is connected. | |||
==Related notions== | ==Related notions== | ||
* [[Quasicomponent]] is a related notion. For a [[locally connected space]] (and for many other kinds of spaces), the quasicomponents coincide with the connected components. In general, each quasicomponent is a union of connected components. | * [[Quasicomponent]] is a related notion. For a [[locally connected space]] (and for many other kinds of spaces), the quasicomponents coincide with the connected components. In general, each quasicomponent is a union of connected components. | ||
Latest revision as of 17:47, 26 January 2012
Definition
Definition as a subset
A connected component of a topological space is defined as a subset satisfying the following two conditions:
- It is a connected subset, i.e., it is a connected space with the subspace topology.
- It is not properly contained in any bigger subset that is connected.
Definition in terms of equivalence relation
For a topological space , consider the following relation: if there exists a subset of containing both and that is a connected space under the subspace topology. Then, it turns out that is an equivalence relation on . The equivalence classes under are termed the connected components of .
The relation is termed the relation of being in the same connected component.
Equivalence of definitions
Further information: equivalence of definitions of connected component
Facts
- The connected components are pairwise disjoint subsets. This follows from the equivalence relation version of the definition.
- Connected components are closed. This follows from the fact that the closure of a connected subset is connected.
Related notions
- Quasicomponent is a related notion. For a locally connected space (and for many other kinds of spaces), the quasicomponents coincide with the connected components. In general, each quasicomponent is a union of connected components.