The continuity of the map lim_T in Hausdorff spaces
  Consider a Hausdorff space (X,T) and a set C of converging nets in X. By virtue of the limit uniqueness, the relation Lim which assigns each member x of X to every net N lying in C that converges to x is a map. Of course, structuring C with some topology U, Lim can be a continuous map. If T is a topology induced by a uniformity, and F is a function space such that X is the codomain of each f in F, it is a well-known property the uniform limits of continuous functions to be continuous. In this paper, the author shows that the continuity of limits of continuous-map nets is implied by the continuity of the map Lim, whenever the involved topologies are large enough, being this result obtained without using neither the uniform convergence notion nor the uniformity concept. 
J. E. Palomar Tarancon