GRE topology question Consider $\mathbb{Z}$ and let $\mathcal{U}$ be the elements of $\mathcal{P}(\mathbb{Z})$ in arithmetic progressions. So $U \in \mathcal{U}$ if there exists $a \in \mathbb{Z}, b \in \mathbb{Z}^+$ such that $$U = \{a+bn : n \in \mathbb{Z}\}.$$ Let $X$ be the integers with $\mathcal{U}$ as a base. Which of the following are true? also if the sequence $n!$ just $\{1,2,6,24,...\}$
(I) $X$ is metrizable.
(II) The sequence $n!$ converges in $X$.
(III) Addition, multiplication, negation are continuous in $X$. (i.e., $X$ is a topological ring).
Thanks in advance. just a tip or hint would be good. My thoughts are that this is a discrete space? even more so than the integers ? So do We get (I) for free right off the bat? I am a bit unsure about the other two.

Just the continuity question.
Write $U_{a,b}=\{a+bn\}$ for a basis element.
That negation is continuous is easy. If $n(x)=-x,$ then $n^{-1}(U_{a,b})=U_{-a,b}.$
Partial proof just of continuity of addition
Let $s(x,y)=x+y.$ Then we need $$s^{-1}(U_{a,b})=\{(x,y)\in\mathbb Z^2\mid x+y\in U\}$$
to be open in the product topology on $\mathbb Z^{2}.$
Show: $$s^{-1}(U_{a,b})=\bigcup_{c\in \mathbb Z}U_{c,b}\times U_{a-c,b}$$
Multiplication is not much harder
Let $p(x,y)=xy.$
Define $$p^{-1}(U_{a,b})=\{(u,v)\mid uv\equiv a\pmod b\}$$
Then show: $$p^{-1}(U_{a,b})=\bigcup_{(u,v)\in p^{-1}(U_{a,b})} U_{u,b}\times U_{v,b}$$
