We have $f(x,y)=xy$ where $f:\mathbb{R}_{[-)} \times \mathbb{R}_{[-)} \to \mathbb{R}$.
What is the quotient topology?
I know that the topological basis in $\mathbb{R}_{[-)} \times \mathbb{R}_{[-)}$ consists of rectangles whose left and bottom are closed but right and top are open. I tried considering $f^{-1}((a,b))$ but that didn't really get me anywhere, and I'm not sure how to proceed.
As $\Bbb S^2$ has a finer topology than the standard Euclidean topology on the plane, and $f$ is Euclidean-Euclidean continuous, $\mathcal{T}_{\text{eucl}} \subseteq \mathcal{T}_q$ (the latter being the quotient topology being induced by $f$).
If putting the lower limit topology on $\Bbb R$ makes $f$ also continuous from the Sorgenfrey plane, then we'd already have that that topology too is a subset of $\mathcal{T}_q$.