Equivalence of definitions of connected component

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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

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:

  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.

Facts used

  1. Connected union of connected subsets is connected

Proof

Proof that the specified relation is an equivalence relation

We prove all three aspects of an equivalence relation:

Condition Statement Proof
reflexivity \! a \sim a, i.e., every point is contained in a connected subset the singleton subset comprising that point is connected
symmetry \! a \sim b \iff b \sim a the definition is symmetric, so this is direct
transitivity \! a \sim b, b \sim c \implies a \sim c From the given, there exists a connected subset containing a and b. Call it D_1. There also exists a connected subset containing b and c. Call it D_2. By Fact (1), D_1 \cup D_2 is connected, and it contains both a and c.

Proof from equivalence relation to subset definition

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

  1. Each equivalence class is a connected subset: This follows essentially from Fact (1). Let C be an equivalence class. Pick any point a in an equivalence class. Then, for any point b in the class, let D(a,b) be a connected subset containing both a and b. The whole equivalence class is a union \bigcup_{b \in C} D(a,b) and hence, by Fact (1), is connected.
  2. 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 C is any maximal connected subset and suppose it is not an equivalence class. Let a \in C and let D be the equivalence class of a. Then, we just noted that D is connected, so C \cup D is connected by Fact (1), hence contradicting the maximality of C.