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. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like
$\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.