Connected component

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Definition

Definition as a subset

A connected component of a topological space is defined as a subset satisfying the following two conditions:

  1. It is a connected subset, i.e., it is a connected space with the subspace topology.
  2. It is not properly contained in any bigger subset that is connected.

Definition in terms of equivalence relation

For a topological space X, consider the following relation: a \sim b if there exists a subset of X containing both a and b that is a connected space under the subspace topology. Then, it turns out that \! \sim is an equivalence relation on X. The equivalence classes under \! \sim are termed the connected components of X.

The relation \! \sim 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.