I said earlier that what convergence was the starting point of topology. Why not take this seriously and replace topological spaces by spaces that would be defined in terms of notions of convergence directly, instead of through opens?
This works well, both with Moore-Smith convergence (in terms of nets) and with filters. One bonus is that we shall obtain Cartesian-closed categories, and this will be easy! Comparing with our exploration of exponential topologies (Section 5.4), of C-generated spaces (Section 5.6), of bc-domains (Section 5.7), this will be a relief.
Oh, just to set it straight: there will be a snag with the net convergence approach. This will force us to use filters in the end. Since I still feel nets are more easily understandable, let’s start with them.
Net convergence spaces.
Let us implement our program of defining a replacement for topological spaces that would be based on convergence directly.
Call a net convergence space any set X together with a relation →, between nets (xi)i ∈ I, ⊑ and points x in X, satisfying the following two axioms, due to Garrett Birkhoff [1, Theorem 7]:
- (4α) If xi=x for every i in I, then (xi)i ∈ I, ⊑ → x
- (4β) If (xi)i ∈ I, ⊑ → x, then (xα (j))j ∈ J, ≤ → x for every subnet (xα (j))j ∈ J, ≤.
The phrase (xi)i ∈ I, ⊑ → x would mean that the net (xi)i ∈ I, ⊑ converges to x, of course. Axiom (4α) states that any constant net converges to the obvious value, while (4β) states that convergence and limits are preserved by taking subnets.
Net convergence spaces form a category, whose morphisms are those maps that preserve convergence. Call such maps continuous: a continuous map is a map f such that if (xi)i ∈ I, ⊑ → x then (f(xi))i ∈ I, ⊑ → f(x). (For those of you who are impatient, the announced snag lies somewhere in this very paragraph.)
Products in this category are defined in the obvious way: a net of tuples converges if and only if it converges componentwise.
Moreover, this category is Cartesian-closed: the exponential object [X → Y] is just the space of continuous maps from X to Y with the structure of continuous convergence… yes, continuous convergence of nets (Section 5.4.1), namely (fi)i ∈ I, ⊑ → f in [X → Y] if and only if for every convergence net (xj)j ∈ J, ≤ → x, (fi(xj))(i,j) ∈ I × J, ⊑ × ≤ → f(x) in Y.
The convergence relation on [X → Y] is in effect defined as the least one that makes App continuous. That’s it.
Much simpler than the situation in topological spaces, right?
But there is a snag, as I said earlier. Have you spotted it?
OK, here it is.
The problem is that nets on a set X do not form a set, but a proper class. This is because there are at least as many nets (xi)i ∈ I, ⊑ on X as there are indexing sets I, and the class of all sets is not a set. In the VBG set theory that I’m using as foundations in the book, you cannot then even define what a relation → would be! A binary relation R between two sets A and B is indeed a subset of A × B. Here we would need to define → as a subset of the binary product of the class of nets with X. But VBG set theory has no provision for forming a product of a proper class with anything: it is just too large.
There are logical ways of fixing this. Use Tarski-Grothendieck set theory (TG), or Morse-Kelley set theory (MK), for example. The latter was even used in topology. But I’d like to stick to a set theory that is as standard as it can be. VBG is a good compromise: as a conservative extension of standard Zermelo-Fraenkel with choice (ZFC) set theory, everything I say in VBG set theory can be recast in ZFC set theory, and VBG gives me the added comfort of being able to talk about classes explicitly. TG and MK are non-conservative extensions, meaning that some statements you can prove in TG or MK are not provable in ZFC.
Filter spaces.
All this is repaired by using filters. The class of filters on a set is a set, getting rid of the problem with nets. And of course, filters are enough to define convergence. We are no longer leaving the cosy realm of standard axioms for Mathematics. The only price we have to pay is to use filter convergence, which is more obscure than net convergence.
OK, let’s redo what we’ve done above, this time in the language of filters.
A filter space is a pair (X, →) where X is a set and → is a relation between filters F and points x, satisfying the following two axioms, analogous to the ones given above for nets:
- (x) → x, where (x) is the filter of all subsets of X that contain x (this is called the principal ultrafilter at x)
- If F → x and F’ is a filter that contains F, then F’ → x.
My reference is Hyland [2], who does wonders with them in the theory of computation. I don’t know who invented them. Variants of these (limit spaces, convergence spaces) already existed before, as one reckons by reading Hyland.
These are the only axioms we are requiring. One could add extra axioms, such as: if F → x and F’ → x then F ∩ F’ → x. This is an example of a property that is true in every topological space, but may fail in filter spaces. Incidentally, filter spaces with this extra property are called convergence spaces.
The category of filter spaces.
Filter spaces form a category, whose morphisms are the continuous maps: a map from a filter space X to a filter space Y is continuous if and only if whenever F → x in X, then f[F] → f(x) in Y. Recall that f[F] = {B ⊆ Y | f–1 (B) is in F} is the image filter of F by f.
One of the nice properties of filter spaces is that they form a Cartesian closed category. This is as for net convergence spaces… but this is really working now.
The binary product X × Y is the usual Cartesian product, with convergence defined by F → (x, y) iff π1[F] → x and π2[F] → y. This is componentwise convergence, as expected. In particular, if F1 → x in X and F2 → y in Y, then F1 ⊗ F2 → (x, y), where F1 ⊗ F2 is the filter of all those sets that contain a rectangle U × V with U in F1 and V in F2: one recognizes the usual pattern defining the product topology.
The exponential object [X → Y] is just the space of continuous maps from X to Y with the structure of continuous convergence… yes, the translation of continuous convergence of nets (Section 5.4.1) into the language of filters: a filter F of continuous maps from X to Y converges continuously to f if and only if, for every filter G that converges to a point x in X, F(G) converges to f(x) in Y. The filter F(G) is just App[F ⊗ G], where App is the usual application map, so the convergence relation on [X → Y] is in effect defined as the least one that makes App continuous. That’s all.
Relation to topological spaces.
Every topological space X can be converted to a filter space Flt(X), by letting → be the usual notion of convergence in X (“F → x iff F contains the neighborhood filter Nx“). Every continuous map f : X → Y between topological spaces defines a map Flt(f) (=f) between filter spaces, which is continuous in the sense that it preserves the convergence relation. This makes Flt a functor.
Conversely, any filter space X, as defined above, defines a topological space Top(X): we let its opens be those subsets U of X such that for every point x in U, for every filter F such that F → x, U is in F. This also defines a functor Top.
Top and Flt are not inverses of each other: in fact there are many more filter spaces than topological spaces. But Top is left adjoint to Flt, so that one can say that Top(X) is the free topological space over the filter space X.
One checks easily that the counit of the adjunction is the identity map: Top(Flt(X)) is just the topological space X. The unit η:X → Flt(Top(X)), where X is a filter space, is more interesting. At the level of points, it is just again the identity map, but more filters converge in Flt(Top(X)) than in X.
Call a filter space topological if and only if η is iso, equivalently, the topological filter spaces are those of the form Top(X) for some topological space. This allows us to consider every topological space X as a topological filter space, equating X with Top(X). But there are more filter spaces than topological spaces.
A quick argument is as follows. Let X be a non-core compact topological space, so that there is a topological space Y such that [X → Y] has no exponential topology. It is an easy exercise to show that the exponential object [Flt(X) → Flt(Y)] in the category of filter spaces cannot be topological.
An example: Scott-convergence.
Let us give another, more concrete example of non-topological filter spaces. These ones will be familiar to domain-theorists (or see Section 4.2). We start with the net convergence view.
Let X be a dcpo. We can define Scott-convergence of nets (xi)i ∈ I, ⊑ in X by (xi)i ∈ I, ⊑ → x if and only if (xi)i ∈ I, ⊑ is a monotone net, and x is below its sup (not equal to its sup: we really mean that x is a limit of (xi)i in I, ⊑ in the Scott topology.) This obeys Birkhoff’s axioms (4α) and (4β). The topology defined by this notion of convergence, which is defined as usual as the collection of subsets such that if (xi)i ∈ I, ⊑ → x ∈ U then xi is in U for i large enough, is exactly the Scott topology. But topological convergence (for the Scott topology) is strictly larger than our original notion of convergence: there are nets that converge topologically, but not with our original definition of →, simply because they are not monotone nets. For example, consider the sequence of intervals [1–1/n, 1–1/n] with n positive natural number in IR, Scott’s domain of closed intervals of the real line ordered by reverse inclusion (Ex. 4.2.28). These intervals can be equated with the reals 1–1/n, which converge to 1 in the usual topology on R, hence to [1, 1] in the Scott topology. But the sequence of these intervals does not converge to anything with respect to →, since it is not a monotone net in IR.
Imitating this with filters, let’s say that F → x (Scott-convergence of filters) if and only if:
- F is the convergence filter of some monotone net, and
- x is below the sup of this monotone net.
Recall that the convergence filter of a net (xi)i ∈ I, ⊑ is the collection of supersets of sets of the form {xi | i ≥ i0} with i0 in I. The second condition can be rewritten as: x is a lower bound of the intersection ⋂A ∈F↑A of the upward closures of elements of F, which shows that it is independent of the chosen monotone net.
As above with nets, the topology defined by this notion of convergence is the Scott topology, so Top(X) really is X with its Scott topology. The notion of convergence in Flt(Top(X)) is larger than Scott-convergence. In IR, the convergence filter F1– of the sequence of intervals [1–1/n, 1–1/n] converges to [1, 1] in the Scott topology, that is, in Flt(Top(X)); but it does not converge to anything in X, because F1– is not the convergence filter of any monotone net.
Topological filter spaces.
Recall that we could equate each topological space X with the filter space Flt(X). Making this identification, the adjunction Top ⊣ Flt can be read by saying that the category of topological spaces is a reflective subcategory of the category of filter spaces.
Can we characterize those filter spaces that are topological? Remember that every topological filter space is a convergence space, meaning that if F → x and F’ → x then F ∩ F’ → x. (By the way, you can check that Scott-convergence of filters, as defined above, does not in general define a structure of a convergence space, giving another argument why Scott-convergence is not topological.) The same property holds even for arbitrary, not just binary, intersections.
In other words, every topological filter space satisfies the much stronger axiom:
- if (Fi)i ∈ I is any family of filters on X such that Fi → x for every i in I, then F → x where F is the intersection of the Fis.
Conversely, any filter space that satisfies the latter axiom is topological. Consider indeed the family of all the filters that converge to x, and take their intersection. It is natural to call this intersection filter Nx, as it will turn out to be the filter of open neighborhoods of x. The extra axiom above implies that Nx → x, and then that a filter converges to x iff it contains Nx. From Nx, we retrieve a topology by declaring open any set U that is a neighborhood of each of its points, i.e., such that U is in Nx for every x in U: in particular, Nx is really the filter of neighborhoods of x: our filter space is indeed topological.
Oops, there is mistake in there! We shall see in Filters, part III where the error lies. Let us have some rest until then.
— Jean Goubault-Larrecq (November 22nd, 2013)
- Garrett Birkhoff. Moore-Smith Convergence in General Topology. The Annals of Mathematics, Second Series, Vol. 38, No. 1 (Jan., 1937), pp. 39-56.
- J. Martin E. Hyland. Filter Spaces and Continuous Functionals. Annals of Mathematical Logic, Vol. 16 (1979), pp. 101-143.