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Abstract. The construction of a free restriction category can be broken into two steps: the construction of a free stable semilattice fibration followed by the construction of a free restriction category for this fibration. Restriction categories produced from such fibrations are “unitary”, in a sense which generalizes that from the theory of inverse semigroups. Characterization theorems for unitary restriction categories are derived. The paper ends with an explicit description of the free restriction category on a directed graph. 1.

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.