Exercise 27.4

Proof. Let $X$ be a connected metric space with the metric $d$, and let $x_0,x_1\in X$ be distinct. Let $d(x_0,x_1)=r$, and define $f(x)=d(x_0,x)$. $f$ is continuous by the discussion on p. 175. We see that $f(x_0)=0,f(x)=r$, and so by the intermediate value theorem (Theorem 24.3), $f(X)\supseteq [0,r]$, i.e., $f$ maps onto $[0,r]$.

Now suppose $X$ is countable. Then, by Theorem $7.1$ there exists a surjective function $g:{\mathbb{Z}}_{\text{+}}\to X$, and so $f\circ g:{\mathbb{Z}}_{\text{+}}\to f(X)$ maps onto $[0,r]$, which is a contradiction since $[0,r]$ is uncountable by Corollary $27.8$. □