# Connected component

From Topospaces

## Contents

## 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.