Lemma 5.7.8. Let $X$ be a topological space. Let $\pi _0(X)$ be the set of connected components of $X$. Let $X \to \pi _0(X)$ be the map which sends $x \in X$ to the connected component of $X$ passing through $x$. Endow $\pi _0(X)$ with the quotient topology. Then $\pi _0(X)$ is a totally disconnected space and any continuous map $X \to Y$ from $X$ to a totally disconnected space $Y$ factors through $\pi _0(X)$.

Proof. By Lemma 5.7.4 the connected components of $\pi _0(X)$ are the singletons. We omit the proof of the second statement. $\square$

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