About the quotient space $Z/p^iZ$.

Question

Fix a prime number $p$ and consider

$Z_p=\{(a_i)_{i\in N}:a_i\in Z/p^iZ_p\ and\ \phi_{i+1}(a_{i+1})=a_i\ for\ every\ i \}\subset \prod_{i=1}^{\infty}(Z/p^iZ)$,

where $\phi_i:Z/p^iZ\rightarrow Z/p^{i-1}Z,i>1$ is the natural projection map.

Endow each $Z/p^iZ$ with the discrete topology and $\prod_{i=1}^{\infty}(Z/p^iZ)$ the product topology.

How to show that $Z_p$ is closed and compact?

I meet this quesion in exercises and just want some hints (solotion better).

Answer 1

$Z_p$ is a closed subset of the compact space $\prod_{i=1}^\infty (Z{/}p^iZ)$ (Tychonoff's theorem: all factors are finite discrete, so compact).

It is posssible to write $Z_p$ as an intersection of closed subsets of the product, and so is closed.