It is clearly closed (indeed, any connected component is closed). If , then is in the same connected component as (since is in the same connected component as and left multiplication by is a homeomorphism), which in turn is in the same connected component as . Using similar reasoning, if is in the connected component as , then is in the same connected component as . Hence is a subgroup.
If is any automorphism of , then . (Indeed, is a connected set containing and therefore . Replacing by its inverse , we similarly have and therefore .) Applying this to inner automorphisms , we conclude that is a normal subgroup of .
Given the fact that is an open surjection, the product is also an open surjection and therefore a quotient map. It follows easily from the universal property of quotient maps that the multiplication therefore descends to a continuous multiplication , so that is a topological group.
Because a topological group is a uniform space, the Hausdorff condition follows from a weaker separation axiom such as (points are closed). It suffices that the identity of be closed. Its complement is the image under of the complement of in (just by examining coset decompositions), which is open. Since is an open map, it follows that is open, so that is closed, as desired.