Ordered Spaces and Quasi-Uniformities on Spaces of Continuous Order-Preserving Functions

In this paper we introduce and investigate the notions of point open order topology, compact open order topology, the order topology of quasi-uniform pointwise convergence and the order topology of quasi-uniform convergence on compacta. We consider the functorial correspondence between function spaces in the categories of topological spaces, bitopological spaces and ordered topological spaces. We obtain extensions to the topological ordered case of classical topological results on function spaces. We also investigate the property of strict complete regularity of these function spaces.