Let $T$ is subset of real numbers .Let $k$ is any number which is not in $T$ .
Define $$kT =\{ kt : t \in T\}. $$ Let $W$ is toplogy(not usual) on $T$ , then can you prove that :
$$H=\{ kU : U \in W \}$$ is topology for $kT$ .
And is the function $f:kT\times kT\to kT$ continuous with respect to $H$ if it is defined as $$f (kt,ks)=k (t+s ),$$ for $t ,s \in T$?
Note that it is given : for every $s,t \in T $, $s+t$ belongs to $T .$
Thew first part follows from the relations $k(\cup U_i)=\cup kU_i$, $k(U \cap V)=kU\cap kV$, $k\emptyset =k\emptyset$. For the second part, taking $T=(1,\infty)$ and $k=1$ you are asking if $(t,s) \to t+s$ is continuous for ANY topology on $(1,\infty)$. This is not true.