nLab
convergence space

Convergence spaces

Idea

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 ConvConv of convergence spaces is a quasitopos and may be thought of as a nice category of spaces that includes Top as a full subcategory.

Definitions

A convergence space is a set SS together with a relation \to from S\mathcal{F}S to SS, where S\mathcal{F}S is the set of filters on SS; if FxF \to x, we say that FF converges to xx or that xx is a limit of FF. This must satisfy some axioms:

  1. Centred: The principal ultrafilter F x={A|xA}F_x = \{ A \;|\; x \in A \} at xx converges to xx;
  2. Isotone: If FGF \subseteq G and FxF \to x, then GxG \to x;
  3. Directed: If FxF \to x and GxG \to x, then some filter contained in the intersection FGF \cap G converges to xx. In light of (2), it follows that FGxF \cap G \to x itself. (Strictly speaking, the relation should not be called directed unless also every point is a limit of some filter, but this follows from 1.)

It follows that FxF \to x if and only if FF xF \cap F_x does. Given that, the convergence relation is defined precisely by specifying, for each point xx, a filter of subfilter?s of the principal ultrafilter at xx. (But that is sort of a tongue twister.)

A filter FF clusters at a point xx if there exists a proper filter GG such that FGF \subseteq G and GxG \to x.

The definition can also be phrased in terms of nets; a net ν\nu converges to xx if and only if its eventuality filter converges to xx.

The morphisms of convergence spaces are the continuous functions; a function ff between convergence spaces is continuous if FxF \to x implies that f(F)f(x)f(F) \to f(x), where f(F)f(F) is the filter generated by the filterbase {F(A)|AF}\{F(A) \;|\; A \in F\}. In this way, convergence spaces form a concrete category ConvConv.

Note that the definition of ‘convergence’ varies in the literature; at the extreme end, one could define it as any relation whatsoever from S\mathcal{F}S (or even from the class of all nets on SS) to SS, but that is so little structure as to be not very useful. Here we follow the terminology of Lowen-Colebunders.

Examples

In measure theory, given a measure space XX and a measurable space YY, the space of almost-everywhere defined measurable functions from XX to YY 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:

  • If FF is a filter such that every proper filter GFG \supseteq F clusters at xx, then FF converges to xx.

Assuming the ultrafilter theorem (a weak version of the axiom of choice), it's enough to require that FF converges to xx whenever every ultrafilter that refines FF converges to xx (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 xx only if it refines (the eventuality filter of) a sequence that converges to xx.)

A pretopological space is a convergence space that is infinitely filtered:

  • If (F α) α(F_\alpha)_\alpha is any family of filters each of which converges to xx, then αF α\bigcap_\alpha F_\alpha converges to xx.

In particular, the intersection of all of the filters converging to xx (the neighbourhood filter of xx) also converges to xx. Note that every pretopological space is pseudotopological.

Any topological space is a convergence space, and in fact a pretopological one: we define FxF \to x if every neighbourhood of xx belongs to FF. A convergence space is topological if it comes from a topology on SS. The full subcategory of ConvConv 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.

Properties

The improper filter (the power set of SS) converges to every point. On the other hand, a convergence space SS is Hausdorff if every proper filter converges to at most one point; then we have a partial function lim\lim from the proper filters on SS to SS. A topological space is Hausdorff in the usual sense if and only if it is Hausdorff as a convergence space.

A convergence space SS 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), SS 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 𝒰SS\mathcal{U}S \to S, where 𝒰S\mathcal{U}S is the set of ultrafilters on SS, such that the composite S𝒰SSS \to \mathcal{U}S \to S is the identity. That is, it is an algebra for the pointed endofunctor 𝒰\mathcal{U}. The compact Hausdorff topological spaces (the compacta) are precisely the algebras for 𝒰\mathcal{U} considered as a monad. If we treat 𝒰\mathcal{U} as a monad on Rel, then the lax algebra?s are the topological spaces in their guise as relational beta-modules.

Topological structure

Given a convergence space, a filter FF star-converges to a point xx if every proper filter that refines FF clusters at xx. (Assuming the ultrafilter theorem, FF star-converges to xx iff every ultrafilter that refines FF converges to xx.) The relation of star convergence makes any convergence space into a pseudotopological space with a weaker convergence. In this way, PsTopPs Top becomes a reflective subcategory of ConvConv over SetSet.

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 UU is a neighbourhood of a point xx if UU belongs to every filter that converges to xx; it follows that UU belongs to every filter that star-converges to xx. 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, PreTopPre Top is a reflective subcategory of ConvConv (and in fact of PsTopPs Top) over SetSet.

Other pretopological notions: The preinterior? of a set AA is the set of all points xx such that AA is a neighbourhood of xx. The preclosure? of AA is the set of all points xx such that every neighbourhood UU of xx meets (has inhabited intersection with) AA. For more on these, see pretopological space.

Given a convergence space, a set GG is open if GG belongs to every filter that converges to any point in GG, or equivalently if GG 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, TopTop is a reflective subcategory of ConvConv (and in fact of PsTopPs Top and PreTopPre Top) over SetSet.

Other topological notions: A set FF is closed if FF meets every neighbourhood of every point that belongs to FF, equivalently if FF equals its preclosure. The interior of AA is the union of all of the open sets contained in AA; it is the largest open set contained in AA. The closure of AA is the intersection of all of the closed sets that contain AA; it is the smallest closed set that contains AA. (For a topological convergence space, the interior and closure match the preinterior and preclosure.)

References

  • Eva Lowen-Colebunders (1989). Function Classes of Cauchy Continuous Maps. Dekker, New York, 1989.

Revised on May 16, 2013 09:51:38 by Urs Schreiber (89.204.130.247)