Bounded complete and dcpo models of T1 spaces

Dongsheng Zhao recently mentioned one recent result of his and Xiaoyong Xi to me, on models of spaces [1], which leads to a curious situation.

Let us call model of a topological space X any poset Y whose subset of Max Y of maximal points, with the subspace topology from the Scott topology on Y, is homeomorphic to X.  This extends the definition in the book (Proposition 7.7.16), where I required Y to be a dcpo.

Clearly, any space that has a model must be T1.  But which T1 spaces exactly do have models?  And which do have models of a particular kind, for example a model that is a dcpo, or a model that is a continuous dcpo, or etc.?

In the book, I mention two theorems by Keye Martin, extending previous results by Jimmie Lawson: the metrizable spaces that have a continuous dcpo model are exactly the completely metrizable spaces (Theorem 7.7.21), and the T3 spaces that have an ω-continuous models are exactly the Polish spaces.

In these pages, I have already mentioned the following theorem by Mummert and Stephan [2]: the countably-based T1 spaces that have an ω-continuous model Y are those that are Choquet-complete.  And there are many more results of that kind.

The curious situation I would like to describe today is the following:

  • Every T1 space has a bounded complete poset model.
  • Every T1 space has a dcpo model.
  • Some T1 spaces have no bounded complete dcpo model, i.e., you cannot have both “bounded complete” and “dcpo” at the same time.

I will develop each point in turn.

Every T1 space has a bounded complete poset model

This is due to Zhao [3] and Erné [4] independently.  We can in fact require the bounded complete poset model to be algebraic.

If X were not just T1 but also sober and locally compact, then we could define a model Y as Q(X), its Smyth powerdomain, consisting of the non-empty compact saturated subsets of X under reverse inclusion (Corollary 8.3.27 in the book).  Now note that by the Hofmann-Mislove theorem (Theorem 8.3.2), Q(X) is isomorphic to the poset of proper Scott-open filters of open subsets of X, when X is sober.  (A filter is proper, or non-trivial, if it does not contain the empty set.)  We may hope of getting rid of the sobriety assumption by moving from Q(X) to proper Scott-open filters.

Zhao’s move to avoid local compactness is to consider the poset of proper, not necessarily Scott-continuous, filters of open subsets of X.  That is fine, but if you do so, and equate the elements x of X with the filters of open neighborhoods of x, then those will not be maximal.  Zhao additionally restricts to those proper filters F such that the intersection of all the elements of F is non-empty.

The poset F(O(X)) of proper filters of open subsets of X is not something new.  It is the subject of Exercise 9.3.10 of the book: F(O(X)) is a Scott domain, that is, an algebraic bc-domain, directed suprema are just unions, and the finite elements of F(O(X)) are the filters ■U, U open in X, where ■U is the filter of all open supersets of U.

Zhao’s poset, call it Zh(X), is the sub-poset of those filters F such that there is a point of X that lies in every element of F.  (This is a slight reformulation of what I announced earlier.)  Zh(X), contrarily to F(O(X)), need not be a dcpo.  However, if a family of filters (Fi)i ∈ I of Zh(X) has a supremum F in Zh(X), then it must be the union ∪i Fi. Indeed, there must be a point x that is in every element of F, hence in every element of every Fi; that shows that ∩i Fi is in Zh(X), and is the required supremum.  In turn, the fact that directed suprema in Zh(X) are computed as in the ambient dcpo F(O(X)) shows that the filters ■U, where U is non-empty and open, are finite in Zh(X), leading us to show that Zh(X) is algebraic.  Finally, it is easy to show that Zh(X) is bounded complete: if F and G are two filters in Zh(X) with an upper bound, then their least upper bound in F(O(X)), which happens to be the set of open sets W that contains the intersection of an element of F and of an element of G, is also in Zh(X): as with the computation of directed suprema, take a point x that is in every element of the chosen upper bound, and realize that it must be in every W constructed as above.

If X is T0, then the space X embeds in Zh(X) through the familiar map η that maps every point x to its set of open neighborhoods.  That is continuous because η-1(↑■U) is just U.  That also shows that η is almost open, and it is injective since X is T0.

If X is T1, then Max Zh(X) consists exactly of the filters of the form η(x), x in X.  Indeed, if F is a maximal element of Zh(X), and x is one of the points that belongs to every element of F, then, first, F must contain all the open neighborhoods of x.  Otherwise, we could add all the intersections of open neighborhoods of x with elements of F to F and still obtain an element of Zh(X).  Then there cannot be two distinct points x and y that both belong to every element of F: since X is T1, take an open neighborhood U of x that does not contain y, and an open neighborhood V of y that does not contain x, then F contains U V, which contains neither point, contradiction.  It follows that F is exactly the set of open neighborhoods of a unique point.

That finishes to show that Zh(X) is a bounded complete, algebraic poset model of X, if X is T1.

Every T1 space has a dcpo model

This is due to Zhao and Xi [5,6], and proceeds by embedding Zh(X) into a dcpo in such a way that no new maximal element is added (and so that the topology on those maximal elements is unchanged, too).

Zhao and Xi provide a general construction to achieve that.  They start by observing that Zh(X) is a bdcpo (as I call them in Proposition 5.1.60 of the book; they call that a local dcpo), namely a poset where every directed family that is bounded from above has a supremum.  Then, they show that for every algebraic bdcpo Y, one can build a new dcpo Z such that Max Y and Max Z are homeomorphic.  Note that Z is a dcpo, not just a bdcpo, but we lose algebraicity in the process.

The construction is as follows: Z is the set of pairs (y, e) of elements of Y where ye and e is maximal in Y, and those pairs are ordered by (y, e) ≤ (y’, e’) if and only if yy’, and either e=e’ or y’=e’.  That ordering is of course very strange, and the proof that Z is indeed a dcpo and that Max Y and Max Z are homeomorphic is a tedious verification.

If Y=Zh(X), then Z consists of pairs (F, x) of a filter F of open sets and of a point that belongs to every element of F.  That is very natural.  But the ordering is somehow weird, and reminiscent of Johnstone’s counterexample (Exercise 5.2.15 in the book).  If you think of elements of Z as being organized in columns, where (F, x) is in column x, then each column x has an element “at infinity”, which you may picture as lying right at the top of the column: this is (η(x), x), writing η(x) for the filter of open neighborhoods of x.  (This is really a maximal element if X is T1.)  To go up from (F, x), you may either go up inside the current column, or you may move directly to the point at infinity of a different column, say x’, where x’ is another point that belongs to every element of F.

The resulting dcpo Z provides a dcpo model for X, for whichever space X, as long as it is T1.  We obtain a dcpo, but we lose algebraicity, and bounded completeness.

Not every T1 space has a model that is both a dcpo and bounded complete

Those constructions tend to suggest that there should be a more elegant construction of a model of X that would be both a dcpo and bounded complete.

That is hopeless, as Zhao and Xi showed [1]: the set N of natural numbers, with the cofinite topology, does not have any bounded complete dcpo model.

At some point, I will need the following result, which I am stating and proving right now, so that we won’t have to take a detour later.

Fact. In a topology space, every directed family is irreducible, in the sense that if it is included in a finite union of closed sets, then it is included in one of them.

Proof. Let D be a directed family.  It is non-empty.  Let C, C’ be two closed sets.  If D is included in CC’, but not in C and not in C’, then there is a point x of D that is not in C, a point x’ of D that is not in C’.  By directedness, there is a point y above both x and x’.  Since closed sets are downwards-closed, y is neither in C nor in C’, hence not in CC’, contradiction.  ☐

Zhao and Xi start by showing some properties of spaces of maximal elements of bounded complete dcpos.  Let Y be a bounded complete dcpo. For every pair of elements y, y’ of Y, say that y and y’ are friends if and only if they have a common upper bound.  (A more usual name would be “compatible” or “coherent”.  The name “friends” is mine, and for the present purpose. Zhao and Xi also use an equivalent definition, but which I find harder to grasp.)

Lemma.  Let Y be a bounded complete dcpo, and y be a point of Y.  The set of friends of y is Scott-closed.

Proof.  It is clearly downwards-closed.  Let (yi)i ∈ I be a directed family of friends of y.  Since Y is a bc-domain, for each i, y and yi not only have an upper bound, but a least upper bound yyi.  The family (yyi)i ∈ I is directed, and its supremum is an upper bound of both y and of supi ∈ I yi.   Hence y and supi ∈ I yi are friends.  ☐

Corollary. Let Y be a bounded complete dcpo, X=Max Y, and y be a point of Y.  Then X ∩ ↑y is closed in X.

Indeed, X ∩ ↑y is just the set of maximal elements of Y that are friends with y.

Now we are ready for Zhao and Xi’s counterexample.

Let N be the set of natural numbers, with the cofinite topology.  (Any infinite set would fit equally well.)  Recall that its closed sets are N itself, plus all the finite subsets of N.

Assume N had a bounded complete dcpo model Y.  Equate N with Max Y.  Let A be the complement of {0} in NA is not closed in N.  Let B=↓A be the downward closure of A in Y.  Since A=NB, B cannot be (Scott-)closed, since otherwise A would be closed in Max Y=N.  Therefore, there is a directed family (yi)i ∈ I in B whose supremum y is not in B.  By definition, y is not below any element of N except 0.  In other words, y is below 0 and, say, not below 1.

Since y is not below 1, some yi is not below 1 either.  (The set of elements below 1, or in general below any fixed element, is Scott-closed.)  Look at the set C=N ∩ ↑yi of friends of yi that are in N.  By the corollary stated above, C is closed in N.  It cannot be the whole of N, because yi is not below 1, which means that 1 is not in C.  Hence C must be finite.  Poor guy, that yi: it only has finitely many natural number friends.

It might even be that yi has no friend at all… or can it?  No, fortunately, yi can count on 0.  Indeed, yi is below y, which is below 0, so yi and 0 are friends.

Now look at the set A’ of its other natural number friends: the finite set of natural numbers above yi, but different from 0.  For every j in I, by directedness there is an yk above both yj and yi; above yk one finds an element a of A, since every yk was taken from B=↓A.  By definition of A, a≠0, and since a is also above yi, a is in A’.  It follows that every yj is in ↓A’.

Remember the fact we proved as a preliminary step: every directed family is irreducible.  Writing ↓A’ as a finite union of closed sets ↓a, a in A’, we obtain that the family (yi)i ∈ I lies entirely inside ↓a, for some a in A’.  Then its supremum y must also be in ↓a.  However, a≠0… but y is not below any element of N except 0: contradiction.

Jean Goubault-Larrecqjgl-2011

 

  1. Dongsheng Zhao and Xiaoyong Xi.  Spaces that have a bounded complete dcpo model.  Rocky Mountain Journal of Mathematics, to appear, 2017.  Note: the copy that D. Zhao gave me is entitled ‘On topological spaces that have a bounded complete dcpo model’.  (Thanks to him!)  It also contains additional results that I have not mentioned here.
  2. Carl Mummert and Frank Stephan.  Topological aspects of poset spaces.  Michigan Mathematical Journal, 59, 2010, pages 3-24.
  3. Dongsheng Zhao.  Poset models of topological spaces.  In Proc. Intl. Conf. Quantitative Logic and Quantification of Software, Global Link publisher, 2009, pages 229-238.
  4. Marcel Erné.  Algebric models for T1 spaces.  Topology and its Applications 158(7), 2011, pages 945-962.
  5. Xiaoyong Xi and Dongsheng Zhao.  Well-filtered spaces and their dcpo models.  Mathematical Structures in Computer Science 27(4), may 2017, pages 507-515. DOI: http://dx.doi.org/10.1017/S0960129515000171
  6. Dongsheng Zhao and Xiaoyong Xi.  Directed complete poset models of T1 spaces.  Mathematical Proceedings of the Cambridge Philosophical Society, october 2016. DOI: https://doi.org/10.1017/S0305004116000888