Reflections in topological algebraic structures

Let C be an epireflective subcategory of Top and let rC be the epireflective functor associated with C. If A denotes a (semi)topological algebraic subcategory of Top, we study when rC(A) is an epireflective subcategory of A. We prove that this is always the case for semi-topological structures and we find some sufficient conditions for topological algebraic structures. We also study when the epireflective functor preserves products, subspaces and other properties. In particular, we solve an open question about the coincidence of epireflections proposed by Echi and Lazaar in [5, Question 1.6] and repeated in [6, Question 1.9]. Finally, we apply our results in different specific topological algebraic structures.