A convergence space is a generalisation of a topological space based on the concept of convergence of filters (or nets) as fundamental. The basic concepts of point-set topology (continuous functions, compact and Hausdorff topological spaces, etc) make sense also for convergence spaces, although not all theorems hold. The category of convergence spaces is a quasitopos and may be thought of as a nice category of spaces that includes Top as a full subcategory.
It follows that if and only if does. Given that, the convergence relation is defined precisely by specifying, for each point , a filter of subfilter?s of the principal ultrafilter at . (But that is sort of a tongue twister.)
A filter clusters at a point if there exists a proper filter such that and .
The morphisms of convergence spaces are the continuous functions; a function between convergence spaces is continuous if implies that , where is the filter generated by the filterbase . In this way, convergence spaces form a concrete category .
Note that the definition of ‘convergence’ varies in the literature; at the extreme end, one could define it as any relation whatsoever from (or even from the class of all nets on ) to , but that is so little structure as to be not very useful. Here we follow the terminology of Lowen-Colebunders.
In measure theory, given a measure space and a measurable space , the space of almost-everywhere defined measurable functions from to becomes a convergence space under convergence almost everywhere?. In general, this convergence space does not fit into any of the examples below.
A pseudotopological space is a convergence space satisfying the star property:
Assuming the ultrafilter theorem (a weak version of the axiom of choice), it's enough to require that converges to whenever every ultrafilter that refines converges to (or clusters there, since these are equivalent for ultrafilters).
A subsequential space is a pseudotopological space that may be defined using only sequences instead of arbitrary nets/filters. (More precisely, a filter converges to only if it refines (the eventuality filter of) a sequence that converges to .)
A pretopological space is a convergence space that is infinitely filtered:
Any topological space is a convergence space, and in fact a pretopological one: we define if every neighbourhood of belongs to . A convergence space is topological if it comes from a topology on . The full subcategory of consisting of the topological convergence spaces is equivalent to the category Top of topological spaces. In this way, the definitions below are all suggested by theorems about topological spaces.
Every Cauchy space is a convergence space.
The improper filter (the power set of ) converges to every point. On the other hand, a convergence space is Hausdorff if every proper filter converges to at most one point; then we have a partial function from the proper filters on to . A topological space is Hausdorff in the usual sense if and only if it is Hausdorff as a convergence space.
A convergence space is compact if every proper filter clusters at some point; that is, every proper filter is contained in a convergent proper filter. Equivalently (assuming the ultrafilter theorem), is compact iff every ultrafilter converges. A topological space is compact in the usual sense if and only if it is compact as a convergence space.
The topological convergence spaces can be characterized as the pseudotopological ones in which the convergence satisfies a certain “associativity” condition. In this way one can (assuming the ultrafilter theorem) think of a topological space as a “generalized multicategory” parametrized by ultrafilters. 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. If we treat as a monad on Rel, then the lax algebra?s are the topological spaces in their guise as relational beta-modules.
Convergence spaces form a full sub-ccc of filter spaces.
Given a convergence space, a filter star-converges to a point if every proper filter that refines clusters at . (Assuming the ultrafilter theorem, star-converges to iff every ultrafilter that refines converges to .) The relation of star convergence makes any convergence space into a pseudotopological space with a weaker convergence. In this way, becomes a reflective subcategory of over .
Note: the term ‘star convergence’ is my own, formed from ‘star property’ above, which I got from HAF. Other possibilities that I can think of: ‘ultraconvergence’, ‘universal convergence’, ‘subconvergence’. —Toby
Given a convergence space, a set is a neighbourhood of a point if belongs to every filter that converges to ; it follows that belongs to every filter that star-converges to . The relation of being a neighbourhood makes any convergence space into a pretopological space, although the pretopological convergence is weaker in general. In this way, is a reflective subcategory of (and in fact of ) over .
Other pretopological notions: The preinterior of a set is the set of all points such that is a neighbourhood of . The preclosure of is the set of all points such that every neighbourhood of meets (has inhabited intersection with) . For more on these, see pretopological space.
Given a convergence space, a set is open if belongs to every filter that converges to any point in , or equivalently if equals its preinterior. The class of open sets makes any convergence space into a topological space, although the topological convergence is weaker in general. In this way, is a reflective subcategory of (and in fact of and ) over .
Other topological notions: A set is closed if meets every neighbourhood of every point that belongs to , equivalently if equals its preclosure. The interior of is the union of all of the open sets contained in ; it is the largest open set contained in . The closure of is the intersection of all of the closed sets that contain ; it is the smallest closed set that contains . (For a topological convergence space, the interior and closure match the preinterior and preclosure.)