Let $\tau_1$ be the cofinite topology over $\mathbb{R}$.
Let's consider the product topology $\tau=\tau_1 \times \tau_1 $over $\mathbb{R}^2$
Let $\tau_2$ be the cofinite topology over $\mathbb{R}^2$.
and $\varepsilon_2 $ the euclidean topology over $\mathbb{R}^2$
Determine if $\tau_2$ and $\varepsilon_2 $ are coarser, finer, the same or not comparable to $\tau$.
Up to now I have shown that che $\tau_1 \times \tau_1 \nsubseteq \tau_2$ by taking $\mathbb{R}\setminus\{0\} \times \mathbb{R}\setminus\{0\} \in \tau_1 \times \tau_1 $ which contains infinite elements and therefore is not the complement of a finite subset. I am trying to show $ \tau_2 \subseteq \tau_1 \times \tau_1 $, but no luck: graphically it seems it is that way, but I am taking an open set $ \tau_2 $ and an element in it and trying to show it belongs to $ \tau_1 \times \tau_1 $, but I am stuck