Exercise 8, § 9

Let $p$ be a prime number. We denote by $Z_n$ the set $Z$ of rational integers, endowed with the inverse image topology of the discrete topology by the canonical mapping of $Z$ onto $Z/p^nZ$; for $m \le n$, we denote by $f_{mn}$ the identical mapping $Z_n \to Z_m$, which is continuous. Now prove that the $Z_n$ are quasi-compact spaces, but that the projective limit of the projective system $(Z_n, f_{mn})$ is not a quasi-compact space.¹