Connected component

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.

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.