nLab convergence space

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 $Conv$ 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 $S$ 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 $x$ or that $x$ is a limit of $F$. 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. Directed: If $F \to x$ and $G \to x$, then some filter contained in the intersection $F \cap G$ converges to $x$. In light of (2), it follows that $F \cap G \to x$ 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 $F \to x$ if and only if $F \cap F_x$ does. Given that, the convergence relation is defined precisely by specifying, for each point $x$, a filter of subfilters? of the principal ultrafilter at $x$. (But that is sort of a tongue twister.)

A filter $F$ clusters at a point $x$, written $F \rightsquigarrow x$, if there exists a proper filter $G$ such that $F \subseteq G$ and $G \to x$.

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 convergence 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, convergence spaces form a concrete category $Conv$.

Note that the definition of ‘convergence’ varies in the literature; at the extreme end, one could define it as any relation whatsoever from $\mathcal{F}S$ (or even from the class of all nets on $S$, see preconvergence space) to $S$, but that is so little structure as to be not very useful. An intermediate notion is that of filter space, in which (3) is not required. Here we follow the terminology of Lowen-Colebunders.

Examples

In measure theory, given a measure space $X$ and a measurable space $Y$, the space of almost-everywhere defined measurable functions from $X$ to $Y$ 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 $F$ is a filter such that every proper filter $G \supseteq F$ clusters at $x$, then $F$ converges to $x$.

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

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

• If $(F_\alpha)_\alpha$ is any family of filters each of which converges to $x$, then $\bigcap_\alpha F_\alpha$ converges to $x$.

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

Any topological space is a convergence space, and in fact a pretopological one: we define $F \to x$ if every neighbourhood of $x$ belongs to $F$. A convergence space is topological if it comes from a topology on $S$. The full subcategory of $Conv$ 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 $S$) converges to every point. On the other hand, a convergence space $S$ is Hausdorff if every proper filter converges to at most one point; then we have a partial function $\lim$ from the proper filters on $S$ to $S$. A topological space is Hausdorff in the usual sense if and only if it is Hausdorff as a convergence space.

A convergence space $S$ 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), $S$ 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 $\mathcal{U}S \to S$, 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. If we treat $\mathcal{U}$ as a monad on Rel, then the lax algebras are the topological spaces in their guise as relational beta-modules.

Topological structure

Given a convergence space, a filter $F$ star-converges to a point $x$, written $F \to^* x$, if every proper filter that refines $F$ clusters at $x$. (Assuming the ultrafilter theorem, $F$ star-converges to $x$ iff every ultrafilter that refines $F$ converges to $x$.) The relation of star convergence makes any convergence space into a pseudotopological space with a weaker convergence. In this way, $Ps Top$ becomes a reflective subcategory of $Conv$ over $Set$.

Note: the term ‘star convergence’ and its symbol ‘$\to^*$’ are 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 $U$ is a neighbourhood of a point $x$, written $x \in^\circ U$, if $U$ belongs to every filter that converges to $x$; it follows that $U$ belongs to every filter that star-converges to $x$. 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, $Pre Top$ is a reflective subcategory of $Conv$ (and in fact of $Ps Top$) over $Set$.

Other pretopological notions: The preinterior of a set $A$ is the set of all points $x$ such that $x \in^\circ A$. The preclosure of $A$ is the set of all points $x$ such that every neighbourhood $U$ of $x$ meets (has inhabited intersection with) $A$. For more on these, see pretopological space.

Given a convergence space, a set $G$ is open if $G$ belongs to every filter that converges to any point in $G$, or equivalently if $G$ belongs to every filter that star-converges to any point in $G$, or equivalently if $G$ 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, $Top$ is a reflective subcategory of $Conv$ (and in fact of $Ps Top$ and $Pre Top$) over $Set$.

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

Theorem

The inclusions $Top \to Pre Top \to Ps Top \to Conv$ are all inclusions of full subcategories over $Set$. That is, they all agree on what a continuous function is.

Proof

The only hard part is proving that, if $f(\mathcal{F}) \to^* f(x)$ whenever $\mathcal{F} \to x$ in a pretopological space, then $x \in^\circ f^*(V)$ whenever $f(x) \in^\circ V$. This is usually proved by contradiction and flagrant use of choice: supposing that $f(x) \in^\circ V$ but $x \notin^\circ f^*(V)$, then every neighbourhood $U$ of $x$ must satisfy $U \nsubseteq f^*(V)$, so choose for each such $U$ a point $y_U$ such that $y_U \in U$ but $f(y_U) \notin V$, defining a net $y$ (indexed by neighbourhoods of $x$ ordered by reverse inclusion), such that $y \to x$, but $\neg\big(f(y) \rightsquigarrow f(x)\big)$, so $y \nrightarrow^* f(x)$, getting a contradiction.

But the theorem is in fact perfectly constructive: the filter $\mathcal{N}_x$ of neighbourhoods of $x$ converges to $x$, so $f(\mathcal{N}_x) \to^* f(x)$; all that really matters is that $f(\mathcal{N}_x) \rightsquigarrow f(x)$, so that for each $V \ni^\circ f(x)$ and $U \ni^\circ x$, for some $W$ with $x \in^\circ W \subseteq U$, $f(W) \subseteq V$, so $W \subseteq f^*(V)$, making $f^*(V)$ a neighbourhood of $x$.

Cluster spaces

The notion of clustering generalizes convergence.

A cluster space is a set $S$ together with a relation $\rightsquigarrow$ from $\mathcal{F}S$ to $S$; if $F \rightsquigarrow x$, we say that $F$ clusters at $x$ or that $x$ is a cluster point of $F$. The axioms are as follows:

1. Centred: The principal ultrafilter $F_x \rightsquigarrow x$;
2. Antitone: If $F \supseteq G$ and $F \rightsquigarrow x$, then $G \rightsquigarrow x$;
3. Codirected: If $F \cap G \rightsquigarrow x$ then $F \rightsquigarrow x$ or $G \rightsquigarrow x$.
4. Nontrivial: If $F \rightsquigarrow x$, then $F$ is proper.

Note that the direction of isotony and directedness for clustering is the reverse of that for convergence (hence ‘antitone’ and ‘codirected’). Nontriviality is the nullary version of directedness (equivalent to the statement that the improper filter never clusters at any point), which we explicitly need this time. Alternatively, we can take $\rightsquigarrow$ as a relation only on the proper filters; then nontriviality may be omitted from the axioms (as was done in the original reference, Muscat 2015).

Every convergence space is a cluster space (using the usual definition of $\rightsquigarrow$ from $\to$), and many of the notions of convergence generalize to cluster spaces, including continuous functions, open/closed sets, neighborhood filters, pre-closure, compactness, etc.

This definition of a cluster space does not seem to work in constructive mathematics. In particular, Muscat's proof that every convergence space satisfies the directedness axiom relies on excluded middle, and the lesser limited principle of omniscience follows if it holds in the real line, for example. It's not clear yet what if any alternative will work better.

References

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

• Joseph Muscat (2015). An axiomatization of filter clustering. Conference: 2015 12th International Conference on Fuzzy Systems and Knowledge Discovery (FSKD). DOI:10.1109/FSKD.2015.7381908.