Semicontinuous limits of nets of continuous functions

In this paper we present a topology on the space of real-valued functions defined on a functionally Hausdorff space X that is finer than the topology of pointwise convergence and for which (1) the closure of the set of continuous functions C(X) is the set of upper semicontinuous functions on X , and (2) the pointwise convergence of a net in C(X) to an upper semicontinuous limit automatically ensures convergence in this finer topology.