On topological systems

We establish the reflectivity of the subcategories of $$T_{0}$$T0 and sober topological systems in the category $$\mathbf {TopSys}$$TopSys of topological systems. We also introduce a Sierpinski object in the category $$\mathbf {TopSys}$$TopSys and point out its connection with $$T_{0}$$T0 and sober topological systems and also with injective $$T_{0}$$T0-topological systems.