I want to show that in $\mathbb{R}^2$ with the cofinite topology, every open set is both connected and path connected.
To do so, I thought of using the following result : for a finite set S, the set $\mathbb{R}^2/S$ is connected in the Euclidean topology.
It gives us that $\forall C \in \mathbb{R}^2 $ open with the cofinite topology ( ie $\mathbb{R}^2 / C $ is finite) , $\mathbb{R}^2/(\mathbb{R}^2 / C) = C $ is connected in the Euclidean topology.
But then I don't see how we could link it with connectedness in the cofinite topology.
Any help would be appreciated. Thank you very much