Asymmetric Topology and Its Applications in Areas of Computer Science

Metrics provide an abstract setting which unifies through axioms the common properties of distances in different contexts. An overarching and a more general concept of continuity becomes possible to define with metric spaces. It is also known that metrics are naturally Hausdorff. The Hausdorff property however loses the partial ordering inherent in $T_0$ spaces, and hence superfluous under Scott's topology. Problems arising in areas of computer science such as denotational semantics and complexity analysis therefore ratified the need to propose metric space generalizations that could be used to define non-Hausdorff topologies. The need for the generalization however pays the price of losing the nicer properties associated with the Hausdorff property. Here, we look into the possibility of weakening to a maximum the imposed conditions required of a self mapping in the postulation of Kleene's fixed point theorem in order to guarantee a fixed point, when the underlying partially ordered set comes from a $T_0$-quasi-metric space. Ultimately, we apply the weakened fixed point technique in asymptotic complexity analysis of algorithms whose running time follows a recurrence equation.