The notion of a partial metric space was introduced by Matthews in 1994 who showed, roughly speaking, how metric–like tools can be extended to non–Hausdorff topologies. He also indicated some applications of this class of spaces in the study of denotational semantics of a programming language.

In this talk we are going to present some necessary and sufficient conditions under which the topology generated by a partial metric is equivalent to the topology generated by a suitably defined metric. Next, we are going to focus on two basic topological properties of partial metric spaces, namely completeness and compactness. In particular, it appears that in these spaces compactness is equivalent to sequential compactness.

Finally, we will focus on a very general fixed point theorem for mappings acting in partial metric spaces. In that theorem we impose some conditions on behavior of considered mappings on orbits and a condition relating orbits of points of small size.