Example of non metrizable topological groups satisfying an extra condition. 
DEFINITION-  A function $f:X \to Y$ is called D-supercontinuous if inverse image of every open set is open $F_{\sigma}$ set.

I am looking for an example of a non metrizable topological group (Hausdorff) in which the group operations are also D-supercontinuous.
Obviously every metric group is such a group but I am having a hard time finding a nontrivial or non metric topological group. I know it must not be second countable.
Any help to point me in the right direction is appreciated. Thanks.
 A: Consider $\mathbb{R}^\infty$ with the weak topology, i.e. the colimit of the sequence of inclusions $\mathbb{R}^0\to\mathbb{R}^1\to\mathbb{R}^2\to\dots$ (concretely, $\mathbb{R}^\infty$ is the set of finite-support sequences of real numbers, with the topology that a set is open iff its intersection with $\mathbb{R}^n$ is open for each $n$).  This is a topological group with respect to coordinatewise addition (this is not obvious--to prove addition is continuous, you have to show $\mathbb{R}^\infty\times\mathbb{R}^\infty$ also has the weak topology; see for instance Theorem A.6 in Hatcher's Algebraic Topology).  It is not first countable and thus not metrizable (again, this is not obvious--you can use an argument similar to the argument in this answer, picking sequences converging to $0$ along each coordinate axis).  However, every open subset of $\mathbb{R}^\infty$ is $F_{\sigma}$: an open set is the union of its intersections with $\mathbb{R}^n$ for each $n$, and the intersection with $\mathbb{R}^n$ is an open subset of $\mathbb{R}^n$ and thus a countable union of closed subsets of $\mathbb{R}^n$, which are then also closed in $\mathbb{R}^\infty$.
