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
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 .
Proof that the specified relation is an equivalence relation
We prove all three aspects of an equivalence relation:
|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 .