nLab pseudotopological space

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

Definitions

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

1. Centred: The principal ultrafilter $F_x = \{ A \;|\; x \in A \}$ at $x$ converges to $x$;
2. Isotone: If $F \subseteq G$ and $F \to x$, then $G \to x$;
3. Star property: If $F$ is a filter such that for every proper filter $G \supseteq F$ there exists a proper filter $H \supseteq G$ with $H \to x$, then $F \to x$.

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 $F \to x$ and $G \to x$, then $F \cap G \to x$.

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

• If $F$ is a filter such that every ultrafilter $U \supseteq F$ converges to $x$, then $F \to x$.

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

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 $F$ clusters at a point $x$ if there exists a proper filter $G$ such that $F \subseteq G$ and $G \to x$; given the ultrafilter principle, we may assume that $G$ is an ultrafilter. Note that an ultrafilter clusters at $x$ iff it converges to $x$.

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

The morphisms of convergence spaces are the continuous functions; a function $f$ between pseudotopological spaces is continuous if $F \to x$ implies that $f(F) \to f(x)$, where $f(F)$ is the filter generated by the filterbase $\{f(A) \;|\; A \in F\}$. In this way, pseudotopological spaces form a concrete category $PsTop$, 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 beta-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 $\mathcal{U}S \to S$, where $\mathcal{U}S$ is the set of ultrafilters on $S$, such that the composite $S \to \mathcal{U}S \to S$ is the identity. That is, it is an algebra for the pointed endofunctor $\mathcal{U}$. The compact Hausdorff topological spaces (the compacta) are precisely the algebras for $\mathcal{U}$ considered as a monad.

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

References

• Mike Shulman (2008); Ultrafilters, Pseudotopological spaces, and Stone-Čech compactification; pdf