# Equivalence of definitions of connected component

This article gives a proof/explanation of the equivalence of multiple definitions for the term Connected component (?)View a complete list of pages giving proofs of equivalence of definitions

## Contents

## Statement

We would like to show that the following definitions of connected component make sense and are equivalent.

### 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 .

## Facts used

## Proof

### Proof that the specified relation is an equivalence relation

We prove all three aspects of an equivalence relation:

Condition | Statement | Proof |
---|---|---|

reflexivity | , i.e., every point is contained in a connected subset | the singleton subset comprising that point is connected |

symmetry | the definition is symmetric, so this is direct | |

transitivity | From the given, there exists a connected subset containing and . Call it . There also exists a connected subset containing and . Call it . By Fact (1), is connected, and it contains both and . |

### Proof from equivalence relation to subset definition

Since is an equivalence relation, its equivalence classes are subsets. We now want to argue that:

- Each equivalence class is a connected subset: This follows essentially from Fact (1). Let be an equivalence class. Pick any point in an equivalence class. Then, for any point in the class, let be a connected subset containing both and . The whole equivalence class is a union and hence, by Fact (1), is connected.
- No equivalence class is contained in a bigger connected subset: If it were, then elements outside the subset would be related to elements inside the subset, contradicting its definition as an equivalence class.

### Proof from subset definition to equivalence relation

For this, suppose is any maximal connected subset and suppose it is *not* an equivalence class. Let and let be the equivalence class of . Then, we just noted that is connected, so is connected by Fact (1), hence contradicting the maximality of .