A pseudotopological space or Choquet space is a generalisation of a topological space based on the concept of convergent ultrafilter as fundamental. This view relies on the ultrafilter theorem to guarantee enough ultrafilters; however, we can also describe a pseudotopological structure in terms of convergence of arbitrary filters satisfying certain properties. In this respect, a pseudotopological space is a special kind of convergence space.
The category of pseudotopological spaces is a quasitopos and may be thought of as a nice category of spaces that includes Top as a full subcategory. It is the minimal quasitopos containing the cartesian closed category of sequential spaces (cf Johnstone's topological topos Thm 10.2). It is also a reflective subcategory of filter spaces (Thm 10.3).
A pseudotopological space is a special case of a convergence space; the star property is a stronger version of the filter property of a convergence space:
The property of isotony gives the converse, so if and only if every ultrafilter refining converges to . Thus a pseudotopology consists precisely of a convergence relation between ultrafilters and points satisfying the single axiom that converges to for every .
As with other convergence spaces, a filter clusters at a point if there exists a proper filter such that and ; given the ultrafilter principle, we may assume that is an ultrafilter. Note that an ultrafilter clusters at iff it converges to .
The morphisms of convergence spaces are the continuous functions; a function between pseudotopological spaces is continuous if implies that , where is the filter generated by the filterbase . In this way, pseudotopological spaces form a concrete category , which is in fact a quasitopos.
The topological spaces can be characterized as the pseudotopological ones in which the convergence satisfies a certain associativity condition; see relational β-module. In this way one can think of a topological space as a multicategory parametrized by ultrafilters; see generalized multicategory.
In particular, note that a compact Hausdorff pseudotopological space is defined by a single function , where is the set of ultrafilters on , such that the composite is the identity. That is, it is an algebra for the pointed endofunctor . The compact Hausdorff topological spaces (the compacta) are precisely the algebras for considered as a monad.
Every pretopological space is also a pseudotopological space; these may be characterised as the infinitely directed pseudotopological spaces.