Other approaches to connected components (from this year's MATH4085 Metric and Topological Spaces)
Hi everyone, In a previous post we looked at a sufficient condition for a union of connected sets to be connected. A relatively easy special case of Lemma 6.12 is the following. Lemma Let \(X\) be a (non-empty) topological space, and let \(E_1\) and \(E_2\) be connected subsets of \(X\) such that \(E_1 \cap E_2 \neq \emptyset\,.\) Then \(E_1 \cup E_2\) is also connected. Armed just with this lemma, we can define an equivalence relation \(\sim\) on a non-empty topological space \((X,\tau)\) as follows: for \(x,y \in X\), \(x\sim y\) if and only if there exists a connected subset \(E\) of \(X\) such that both \(x\) and \(y\) are elements of \(E\). Let's check that this really is an equivalence relation. Since single point sets \(\{x\}\) are connected, reflexivity is easy. Symmetry is also immediate (it doesn't matter in which order you say that both \(x\) and \(y\) are elements of \(E\)). That leaves transitivity, which is where we use the lemma above. Let \(x,y,z \in X\) and ...