My question is regarding the existance of "countably close" open sets to those of a parotopological group (or any group equipped with a topology).
Consider a group $G$ equipped with a certain topology $\tau$ making the space $G_{\tau}$.
Is there another topology $\tau'$ on $G$ making the space $G_{\tau'}$ such that the difference between each open set $U\subseteq G_{\tau'}$ and some open set $V\subseteq G_{\tau}$ is countable $|U\setminus V|\leq\aleph_{0}$
I tried to find a justification for the existance of $G_{\tau'}$ by showing that the three axioms of a topology are preserved.
A Simple example is that of the Euclidean (real) line $\mathbb{R}$ and the Sorgenfrey line $\mathbb{S}$ in which each open set $[a,b)$ has a coressponding $(a,b)$ so that their difference is just one element.