= Solution
For each $k$, the coordinate projection
$$
\pi_k:X\to\mathbb Z,
\qquad
\pi_k((x_n))=x_k,
$$
is continuous when $\mathbb Z$ has the <discrete topology>: if $d(x,y)<2^{-k}$, then $x_k=y_k$.
If $\gamma:[0,1]\to X$ were a <continuous path>, then $\pi_k\circ\gamma$ would be a continuous map from the connected interval $[0,1]$ to a discrete space, so it would be constant. This holds for every $k$, and therefore $\gamma$ itself must be constant. Since $X$ contains distinct points, no path joins every pair. Hence
$$
\boxed{(X,d)\text{ is not path connected};}
$$
indeed, every path component is a singleton.
Back to article page
