Assume that $(\mathbb Z,+)$ be group which has been armed by topology
$T=\{ \mathbb Z,\emptyset \} \cup\{[n,\infty), n \in \mathbb Z\}$
How to show that additive function($(a,b)\to_f a+b)$ is is continuous under defined topology and inverse function( $a\to_ga^{-1})$ is not continuous
since pull back of open set $[n,\infty)$ under $g$is not open then $g$ is no not continuous
I tried to use $\epsilon, \delta $ definition to prove continuity of $f$
$$\forall \quad U_{a+b} : \exists \quad U_a,U_b \quad\text{s.t}\quad U_a+U_b\subset U_{a+b} $$
since all open set in T has form of $[n,\infty)$ or$[m,n)$ enough to show that
above statement for all for every all possible open set but I think its rough way for this problem. Does the easy solution??
For $n\in\Bbb Z$ let $U_n=[n,\to)$. Then
$$f^{-1}[U_n]=\{\langle k,\ell\rangle\in\Bbb Z\times\Bbb Z:k+\ell\ge n\}\;,$$
so you need to show that this set is open for any $n\in\Bbb Z$.
HINT: Given $\langle k,\ell\rangle\in f^{-1}[U_n]$, consider the set $U_k\times U_\ell$. Sketching $\Bbb Z\times\Bbb Z$ and $U_n$ might help.