An axiomatization of filter clustering

Smyth, Abramsky, and Vickers have been foremost in presenting topological spaces in computer science as a particular form of fuzzy logic that captures the idea of observations and semi-decidable statements. This article argues for a more general type of space that focuses on the concept of a filter, seen as a gathering of information or refinement process. A new category of cluster spaces is introduced, defined as sets having a clustering function that maps filters to their subsets of cluster points. Axioms for these spaces are formulated and shown to include topological spaces and convergence spaces as natural special cases.