nLab
pseudotopological space

Pseudotopological spaces

Idea

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

Definitions

A pseudotopological 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. This must satisfy some axioms:

  1. Centred: The principal ultrafilter F x={AxA}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. Star property: If FF is a filter such that for every proper filter GFG \supseteq F there exists a proper filter HGH \supseteq G with HxH \to x, then FxF \to x.

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:

  • If FxF \to x and GxG \to x, then FGxF \cap G \to x.

Assuming the ultrafilter principle (a weak version of the axiom of choice), the star property can be expressed in terms of ultrafilters:

  • If FF is a filter such that every ultrafilter UFU \supseteq F converges to xx, then FxF \to x.

The property of isotony gives the converse, so FxF \to x if and only if every ultrafilter refining FF converges to xx. Thus a pseudotopology consists precisely of a convergence relation between ultrafilters and points satisfying the single axiom that F xF_x converges to xx for every xx.

A subsequential space is a pseudotopological space that may be defined using only sequences instead of arbitrary nets/filters.

As with other convergence spaces, a filter FF clusters at a point xx if there exists a proper filter GG such that FGF \subseteq G and GxG \to x; given the ultrafilter principle, we may assume that GG is an ultrafilter. Note that an ultrafilter clusters at xx iff it converges to xx.

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 pseudotopological 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, pseudotopological spaces form a concrete category PsTopPsTop, which is in fact a quasitopos.

Properties

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 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.

Every pretopological space is also a pseudotopological space; these may be characterised as the infinitely directed pseudotopological spaces.

References

Revised on February 2, 2014 01:43:45 by Rasmus Bentmann (87.178.210.113)