Exercise 23.8

Solution. Let ${ℝ}^{\omega }=A\cup B$, where $A$ is the set of bounded sequences and $B$ is the set of unbounded sequences of reals. $A,B$ are disjoint, and so it remains to show they are open. Suppose $a=(a_1,a_2,\dots )\in A$ and $b=(b_1,b_2,\dots )\in B$. Since $\left|a_i\right|<N$ for all $i$ for some $N$, and since $\left|b_i\right|>N+1$ for all $i$ larger than some $I$, we have that $\overline{d}(a_i,b_i)=1$ for all $i\ge I$. Thus, $\overline{\rho }(a,b)=1$ for any $a\in A,b\in B$, and so the open balls with radius $1∕2$ around $a,b$ are fully contained in $A,B$ respectively. □