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
- 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.