Concrete functors that respect initiality and finality

We study concrete endofunctors of the category of convergence spaces and continuous maps that send initial maps to initial maps or final maps to final maps. The former phenomenon turns out to be fairly common while the latter is rare. In particular, it is shown that the pretopological modification is the coarsest hereditary modifier finer than the topological modifier and this is applied to give a structural interpretation of the role of Fréchet-Urysohn spaces with respect to sequential spaces and of k' -spaces with respect to k -spaces.