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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 ...