Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 1 / 12G (Metric and Topological Spaces)

Nonnegativity and symmetry are immediate. Moreover, $d(x,y)=0$ exactly when $n_{\min }(x,y)=\infty$, which means $x_n=y_n$ for every $n$.

For three sequences $x,y,z$, if $x$ and $y$ agree through coordinate $r-1$ and $y$ and $z$ agree through coordinate $s-1$, then $x$ and $z$ agree through coordinate $\min (r,s)-1$. Henceand thereforeThus $d$ is a metric; in fact, the stronger first inequality makes it an ultrametric.

A metric space is complete when every Cauchy sequence converges to a point of the space.

Let $x^{(1)},x^{(2)},\dots$ be a Cauchy sequence in $X$, where $x^{(j)}=(x_1^{(j)},x_2^{(j)},\dots )$. For each positive integer $N$, choose $J_N$ such thatThis inequality says that the first $N$ coordinates of $x^{(j)}$ and $x^{(k)}$ agree. Consequently each coordinate $x_n^{(j)}$ is eventually constant. Let $x_n$ be its eventual value and put $x=(x_1,x_2,\dots )\in X$.

For $j\ge J_N$, the first $N$ coordinates of $x^{(j)}$ agree with those of $x$, soAs $N$ is arbitrary, $x^{(j)}\to x$. Therefore

For each $k$, the coordinate projectionis 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. Henceindeed, every path component is a singleton.