Connected component

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.