Narrow Results

We show that a whole bunch of well known topological constructs have isomorphic descriptions by means of sets structured by some collection of (generalized) metrics of a certain type together with suitable morphisms. The classification of the categories listed in the table at the end of the paper, is based on different saturation conditions acting on certain collections of generalized metrics. These conditions are composed out of different building blocks. For instance on collections of quasi-pre-metrics we encounter saturation conditions that are combinations of the following properties: prime ideal, ideal, locally qualified saturation, uniform qualified saturation, locally quantified saturation and uniform quantified saturation. Combinations of these enable us to characterize the constructs in the first colum of the summarizing table at the end of the paper: pretopological spaces, preclosure spaces, semi-quasi uniform spaces, pre-approach spaces, quasi -pre metric spaces. The more important subconstructs of these examples, such as topological spaces, completely regular spaces, uniform spaces, proximity spaces, approach spaces and metric spaces, are characterized using essentially the same building blocks of saturation properties, restricted to particular subclasses of generalised metrics.