The complete lattice Lfan (part II)

Last time, we had started to study the following poset L_fan, introduced independently in [3, 1, 4]: L_fan is N × N, with equality as ordering on the first component and the usual ordering on the second component, plus an extra bottom element ⊥, plus an extra top element ⊤:

We had seen that L_fan is a complete lattice with the Chen-Kou-Lyu property—in particular, (L_fan × L_fan)_σ = (L_fan)_σ × (L_fan)_σ, where the σ subscript means “with the Scott topology”—although it is neither core-compact nor first-countable (in its Scott topology).

The Scott topology on L_fan can be described explicitly. For every function f : N → N, let U_f be the subset of those pairs (m, n) of N × N such that n≥f(m), plus ⊤.

Then the Scott-open subsets of L_fan are exactly the sets U_f, where f ranges over the functions from N to N, plus the empty set, plus the whole space L_fan. Additionally, U_f ⊆ U_g if and only if f≥g, so that O(L_fan) is order-isomorphic to the poset (N → N)^op of functions from N to N, ordered with the opposite ordering ≥ (not ≤), with an additional bottom element (∅) and an additional top element (L_fan).

Sobriety

We start this month’s post with a nice piece of news about L_fan.

Lemma A. L_fan is sober in its Scott topology.

Proof. Let C be an irreducible subset of L_fan. It is non-empty, and downwards-closed, hence contains ⊥. If it only contains ⊥, then C is the downward closure of ⊥. If C contains ⊤, then C = ↓⊤. Henceforth, let us assume that C is neither of those. But C is closed, the closed sets are the complements of open sets, and we know what the open sets are: C is the complement of U_f, for some function f from N to N. Also, f is not the constant 0 function, since otherwise C would be {⊥}.

For every m ∈ N, let us say that C has a non-trivial column, and precisely that it has non-trivial column number m if and only if C contains a point (m, n) for some n, equivalently if and only if f(m)≠0. If so, there is a largest point (m, n) in C’s column m, namely (m, f(m)–1). Since f is not the constant 0 function, C has at least one non-trivial column. We claim that it has exactly one.

Otherwise, let us say that columns m and m’ are non-trivial, with m≠m’. Let g : N → N be as f, except that g maps m to f(m)–1, and let g’ be as f except that it maps m’ to f(m’)–1. Then C intersects U_g at (m, f(m)–1), U_g’ at (m’, f(m’)–1), but C does not intersect U_g ∩ U_g’, since the latter is equal to U_sup(g,g’) = U_f. This contradicts the fact that C is irreducible (see Trick 8.2.3 in the book).

Therefore C has exactly one non-trivial column, say with number m. Then C = ↓(m, f(m)–1). In all cases, C is the downward-closure of a (unique) point, so L_fan is sober. ☐

This was proved by Chen, Kou and Lyu [1, Example 4.13].

Weak Hausdorffness, and being a maximal limit space

The following properties were not considered by Chen, Kou and Lyu [1], or by Hertling [4], or by Xu, Shen, Xi and Zhao [3], but I think they deserve to be mentioned, specially considering that they were the topic of some of my recent posts.

A space X is weakly Hausdorff in the sense of Klaus Keimel and Jimmie Lawson [5] if and only if for any two points x and y in X, for every open neighborhood W of ↑x ∩ ↑y, there is an open neighborhood U of x and there is an open neighborhood V of y such that U ∩ V ⊆ W. I have already written a few times about that notion.

Lemma B. L_fan is weakly Hausdorff in its Scott topology.

Proof. We take two points x and y and an open neighborhood W of ↑x ∩ ↑y. The existence of U and V is obvious if x≤y (or symmetrically if y≤x): simply take the whole space for U and W for V. Hence let us assume that x and y are incomparable. Then none of x or y is equal to ⊤ or to ⊥, so we can write x as (m, n) and y as (m’, n’), where additionally m≠m’.

The intersection ↑x ∩ ↑y is then equal to {⊤}. If W is the whole space L_fan, then we can simply take U and V equal to the whole space as well. Henceforth, let us assume that W is a proper open subset of L_fan. Since W contains {⊤}, it is non-empty, hence of the form U_f, for some function f : N → N, by Lemma E. We let g : N → N be defined as f, except that it maps m to 0; and g’ be defined as f except that it maps m’ to 0. Then x = (m, n) is in U_g, y = (m’, n’) is in U_g’, and U_g ∩ U_g’ = U_sup(g,g’) = U_f = W, so we can take U ≝ U_g and V ≝ U_g’. ☐

In this post (March 2024) on Hoover’s maximal limit spaces [6], namely on spaces in which every convergent filter has a unique largest limit, we have shown (Theorem I there) that a maximal limit space is the same thing as a weakly Hausdorff, binary-bounded-sup-complete, monotone convergence space. By Lemma A, L_fan is sober, hence a monotone convergence space (see Proposition 8.2.34 in the book), it is weakly Hausdorff by Lemma B, and it is certainly binary-bounded-sup-complete, since it is a complete lattice, as we have seen last time. Therefore:

Proposition C. L_fan is a maximal limit space.

In particular, it is coherent, too. We will retrieve this below, when we study the compact saturated subsets of L_fan.

The dcpo O(L_fan) of Scott-open subsets of L_fan

The complete lattice O(L_fan) of Scott-open subsets of L_fan consist of the sets U_f, with f : N → N, ordered with the opposite of the pointwise ordering on functions f, plus a top element (the whole space L_fan), plus a bottom element (the empty set).

Let us write N_ω for the set of natural numbers N, with an extra element ω above all natural numbers. We order it in the usual way: 0 ≤ 1 ≤ … ≤ n ≤ … ≤ ω.

For every function g from N to N_ω, let dom g be the set of numbers m ∈ N such that g(m)≠ω. This is the domain of g, which I will write as dom g. We let FF be the set of functions f from N to N_ω whose domain is finite, ordered pointwise. Hertling [4, before Lemma 4.13] uses actual total functions from a finite subset of N to N (not N_ω), but I think this definition of FF will be more practical.

For every g ∈ FF, let V_g be the collection of Scott-open subsets of L_fan of the form U_f with f≤g, plus L_fan itself. The following is due to Hertling [4, Lemma 4.14].

Proposition D. The sets V_g, g ∈ FF, plus {L_fan} and O(L_fan), form a base of the Scott topology on O(L_fan).

Proof. We first check that V_g is Scott-open. It is clear that V_g is upwards-closed; notably, if U_f is in V_g, then f≤g, so every U_f’ containing U_f must be such that f’≤f (that was Fact F of last time), whence U_f’ is in V_g, too. We consider any directed family D of Scott-open subsets of L_fan whose supremum is in V_g, and we wish to show that D intersects V_g. If L_fan ∈ D, then some element of D (L_fan itself) is in V_g, and we are done. Hence we assume that L_fan is not in D. It cannot be the case that D only contains the empty set, since sup D would be the empty set, and ∅ is not in V_g. Then we may assume that D does not contain the empty set, without loss of generality: removing the empty set yields a cofinal subfamily, which will in particular be directed and have the same supremum. This way, we may assume that D is a directed family of sets of the form U_fi, i ∈ I. Using the order-isomorphism with (N → N)^op plus extra top and bottom elements (Fact F of last time), its supremum is U_f where f is the pointwise infimum of (f_i)_{i ∈ I}.

By assumption, U_f is in V_g, so f≤g. For each one of the elements m of the domain of g, we can then find an index i ∈ I such that f_i(m)≤g(m); indeed, the family (f_i(m))_{i ∈ I} is well-founded, namely it reaches its least element. Since dom g is finite and (f_i)_{i ∈ I} is directed (with respect to ≥), we can take the same i ∈ I for every m ∈ dom g; namely, for every m ∈ dom g, f_i(m)≤g(m). For every natural number m outside dom g, we have g(m)=ω, so f_i(m)≤g(m) as well. Therefore f_i≤g. This terminates our argument that V_g is Scott-open.

Second, O(L_fan) is trivially Scott-open, and {L_fan} is Scott-open, too: given any directed family of Scott-open subsets (U_i)_{i ∈ I} of L_fan whose union is in {L_fan}, namely whose union is equal to L_fan, or equivalently, whose union contains ⊥, some U_i must contain ⊥ already, namely U_i=L_fan.

Finally, we need to show that the set V_g, {L_fan} and O(L_fan) form a base of of the Scott topology on O(L_fan). Let U ∈ O(L_fan), and let W be a Scott-open subset of O(L_fan) such that U ∈ W. If U=L_fan, then U ∈ {L_fan} ⊆ W. If U=∅, then W=O(L_fan). It remains to see what happens when U = U_f for some function f from N to N. For every k ∈ N, let g_k : N → N_ω map every natural number m<k to f(m), and all others to ω. Clearly, U = U_f ∈ V_gk for every k ∈ N (since f≤g_k), and we claim that V_gk ⊆ W for some k ∈ N; this will finish the proof.

Let us assume that V_gk is not included in W for any k ∈ N. For every k ∈ N, there is an element of V_gk that is not in W. That element cannot be L_fan, since L_fan is in W (W is non-empty since it contains U, and upwards-closed, hence contains the largest element of O(L_fan)). It cannot be the empty set either, since the empty set is not in V_gk. Hence it must be equal to U_fk for some function f_k : N → N. For the record, we will to remember that: (a) U_fk ∈ V_gk and: (b) U_fk ∉ W.

Let f’ : N → N map every number m to max(f(m), f₀(m), …, f_m(m)). Then f≤f’, and: (c) f_k(m)≤f’(m) for every m≥k.

For every k ∈ N, we now define f’_k : N → N as mapping every number m<k to f(m) and every number m≥k to f’(m). Then f’ = f’₀ ≥ f’₁ ≥ … ≥ f’_k ≥ …, and inf_{k ∈ N} f’_k = f. Using the order-isomorphism with (N → N)^op plus extra top and bottom elements (Fact F of last time), U_f’ = U_f’0 ⊆ U_f’1 ⊆ … ⊆ U_f’k ⊆ …, and sup_{k ∈ N} U_f’k = U_f. Since the latter is in W, which is Scott-open, some U_f’k is in W.

We claim that f’_k ≥ f_k. For every m≥k, f’_k(m) = f’(m) ≥ f_k(m), by (c) (see a few lines back). For every m<k, f’_k(m) = f(m); since U_fk is in V_gk (by (a)), f_k≤g_k, so f_k(m)≤g_k(m), but since m<k and by definition of g_k, g_k(m)=f(m); so f’_k(m) = f(m) = g_k(m) ≥ f_k(m).

We can now conclude: f’_k ≥ f_k, so by the usual isomorphism (Fact F of last time), U_f’k is included in U_fk; since U_f’k is in W and W is upwards-closed, U_fk is in W, too. But that contradicts the definition of U_fk, namely (b). Therefore V_gk ⊆ W for some k ∈ N, as promised. ☐

As a side remark, the base given in Proposition D is countable. Therefore, and while L_fan is not first-countable hence not second-countable, O(L_fan) is second-countable in its own Scott topology, as was noted by Hertling.

L_fan is consonant

A space X is consonant if and only if for every open subset U of X, for every Scott-open neighborhood W of U in OX, there is a compact saturated subset Q of X such that Q ⊆ U and every open neighborhood of Q in X belongs to W. I have talked quite a few times about that property. Let me recall that every locally compact space is consonant (Exercise 5.4.12 in the book), and that every regular Čech-complete space is consonant (the Dolecki-Greco-Lechicki theorem, see Exercise 8.3.4 in the book).

Lemma E. Every sober space X such that OX has a base of Scott-open subsets U that are filters (viz., [upwards-closed, non-empty and] such that the intersection of any two elements of U is in U) is consonant.

Proof. Let U be an open subset of X, and W be a Scott-open neighborhood of U in OX. By assumption, there is a Scott-open neighborhood U of U included in W that is a filter. By the Hofmann-Mislove theorem (Theorem 8.3.2 in the book), since X is sober, U is the collection of open neighborhoods of some compact saturated subset Q of X. This is the desired set Q. ☐

The sets V_g, with g ∈ FF, are filters. Given any two open subsets in V_g, if one is equal to the whole of L_fan, then their intersection is the other one, which is in V_g; otherwise, they are of the form U_f and U_f‘ for some functions f, f’ : N → N such that f, f’ ≤ g, and by the order-isomorphism with (N → N)^op plus a top and a bottom element (Fact F of last time), their intersection is U_max(f,f’); since max(f,f’)≤g, U_max(f,f’) is also in V_g. It is easy to see that {L_fan} and O(L_fan) are filters, too. Those form a base of the Scott topology on O(L_fan), by Proposition D. Since L_fan is sober (Lemma A), we can apply Lemma E, and we obtain the following.

Proposition F. L_fan is consonant in its Scott topology.

The compact saturated subsets of L_fan

Given any function g ∈ FF, let Q_g be the subset of L_fan consisting of ⊤ plus all pairs (m, n) where m ∈ dom g and n≥g(m). Hence Q_g is defined just like the Scott-open set U_f, except that g is a slightly different kind of function; also, while both Q_g and V_g are indexed by the same kind of functions g, V_g was a Scott-open subset of O(L_fan), while Q_g will be a compact saturated subset of L_fan. There is a link, though: V_g is the collection of open neighborhoods of Q_g in L_fan.

Here is an illustration of what Q_g looks like, with a function g of domain {0, 2}:

Lemma G. The compact saturated subsets of L_fan are the empty set, the whole space L_fan and the sets of the form Q_g, where g ranges over FF. They are all finitary compact, namely, upward closures of finite sets.

Proof. All those sets—the empty set, L_fan, Q_g—are finitary compact; notably, Q_g is the upward closure of the finite set of points (m, g(m)), m ∈ dom g if dom g is non-empty, and is the upward closure of ⊤ if dom g is empty. Therefore they are all upwards-closed and compact.

Conversely, let Q be a non-empty proper compact saturated subset of L_fan. Being proper, it does not contain ⊥. Since it is non-empty, it contains ⊤. For every m ∈ N, let us say that column m is empty if and only if Q contains no pair (m, n) for any n (but for this particular value of m); it is non-empty otherwise.

Let I be the set of indices m ∈ N such that column m is non-empty. There is a smallest natural number n such that (m, n) is in Q, and we call it g(m). For every number m outside I, we let g(m) ≝ ω. This defines a function g from N to N_ω, whose domain is I.

By way of contradiction, let us assume that I is infinite. We build a function f_m : N → N, one for each m ∈ I, which maps m to g(m), every m’ ∈ I–{m} to g(m’)+1, and all other indices to some arbitrary value, say 0. The Scott-open sets U_fm, m ∈ I, form an open cover of Q: every point of Q (other than ⊤, which is in any U_fm anyway—and there is such a U_fm, since I is non-empty) is of the form (m, n) with n≥g(m)=f_m(m), hence is in U_fm. That open cover has no strict subcover: if we remove any set U_fm (m ∈ I) from the cover, the point (m, f(m)) of Q fails to be in any U_fm’, m’ ∈ I–{m}. Since I is infinite, this contradicts the fact that Q is compact.

Hence I is finite. Therefore g is in FF, and it is easy to see that Q=Q_g. ☐

Fact H. For all g, g’ ∈ FF, Q_g ⊇ Q_g’ if and only if g ≤ g’.

With this characterization, we retrieve some results we have already obtained, in a more concrete fashion:

• L_fan is not core-compact. Indeed, L_fan is sober (Lemma A) and in that case, core-compactness is equivalent to local compactness (see Theorem 8.3.10 in the book). But the only compact saturated subset of L_fan with non-empty interior is L_fan itself. Indeed, the proper compact saturated subsets are of the form Q_g, and none contains any Scott-open subset of the form U_f. Explicitly, U_f ⊆ Q_g if and only if f ≥ g, and that cannot happen for an N-valued function f and a function g in FF.
• L_fan is coherent. This is a consequence of the fact that L_fan is a maximal limit space, which we have seen in Proposition C. More directly, the intersection of Q_g and of Q_g’ is equal to Q_h, where h maps every m ∈ N to max(g(m), g’(m)).

The Smyth hyperspace of L_fan

Last time (Corollary C), we had mentioned that, because L_fan has the Chen-Kou-Lyu property, the Scott topology coincides with the lower Vietoris topology on the Hoare hyperspace H(L_fan). Having explored the shape of compact saturated subsets of L_fan, we can now investigate the other classical hyperspace, the Smyth hyperspace.

The Smyth hyperspace QX of a space X is the space of all compact saturated subsets of X, with the upper Vietoris topology: its basic open subsets are of the form ☐U ≝ {Q ∈ QX | Q ⊆ U}, where U ranges over the open subsets of X.  QX is a T₀ space, whose specialization ordering is reverse inclusion. With what we know about the compact saturated subsets of L_fan, we can show the following.

Proposition I. The upper Vietoris topology coincides with the Scott topology of ⊇ on Q(L_fan) (where L_fan is given its Scott topology).

Proof. First, every upper Vietoris open subsets of Q(L_fan) is Scott-open. Indeed, since L_fan is sober (Lemma A), hence well-filtered (Proposition 8.3.5 in the book), the supremum of any directed family (Q_i)_i∈I in Q(L_fan) is its intersection ∩_i∈I Q_i (Proposition 8.3.25 in the book), and ∩_i∈I Q_i is in a basic open set ☐U if and only if some Q_i is in ☐U already (this is well-filteredness).

Second, let U be any Scott-open subset of Q(L_fan). For every Q ∈ U, we will find a Scott-open subset V of L_fan such that Q ∈ ☐V ⊆ U. By Lemma G, Q may have one of three possible forms. If Q=L_fan, then U is the whole of Q(L_fan), since U is upwards-closed and Q is the least element of Q(L_fan); then we may define V as L_fan. If Q is empty, then we may take V ≝ ∅. In the case that remains, Q = Q_g for some unique g ∈ FF, and we build a sequence of functions g_n in FF by induction on n ∈ N, as follows. The invariants we will maintain are:

• (A) Q_gn is in U;
• (B) the domain of g_n+1 is equal to the domain of g_n plus the one-element set {k_n}, where (k_n)_n∈N is a fixed enumeration without repetition of those natural numbers that lie outside the domain of g.

We start by defining g₀ as g. Given that we have defined g_n, we observe that the functions g_n[k_n ↦ m], defined just as g_n except that k_n (which is outside the domain of g_n) is mapped to the natural number m, form a monotone sequence when m varies over N, and that their supremum is g_n. Through the order-isomorphism stated in Fact H, Q_gn is the supremum of the monotone sequence of compact saturated sets Q_{gn[kn ↦ m]}, m ∈ N. Since Q_gn is in U and U is Scott-open, Q_{gn[kn ↦ m]} is in U for some m ∈ N. We pick one such m, for example the least one, and we define g_n+1 as g_n[k_n ↦ m]. Hence Q_gn+1 is in U (see (A)) and (B) is true.

Now that we have defined the functions g_n, we define f : N → N as the pointwise infimum of g_n, n ∈ N. There is a simpler way of describing f: for every m ∈ N, g_n(m) may be equal to ω for small values of n, but for any n large enough, g_n(m) will be the same natural number, independently of n; and that number is f(m). This is a consequence of (B).

We let V ≝ U_f. This is Scott-open in L_fan. Also, Q = Q_g is included in U_f, hence is an element of ☐V. Indeed, ⊤ is in U_f, and all the other elements of Q_g are pairs (m, n) of natural numbers with n≥g(m); since n is a natural number, g(m)≠ω, so m is in the domain of g=g₀, and therefore g(m)=f(m), whence n≥f(m).

Finally, we claim that ☐V ⊆ U. Let Q’ be any element of ☐V. Q’ cannot be the whole of L_fan, since otherwise ⊥ would be in Q’, hence in V = U_f, and that is impossible. If Q’ is empty, then certainly it is in U, because U is upwards-closed, non-empty (U contains Q) and therefore contains the top element of Q(L_fan), which is the empty set. Finally, and using Lemma G, the only case that remains is that Q’ would be of the form Q_g’ for some function g’ in FF. Since Q’ = Q_g’ ⊆ V = U_f, every pair (m, n) such that n≥g’(m) satisfies n≥f(m), so g’≥f. Since the domain dom g’ of g’ is finite, and since the domains of the functions g_n form a monotone sequence of sets dom g ∪ {k₀, …, k_n–1} as n varies, and whose union is the whole of N, it must be the case that dom g’ ⊆ dom g_n for some n large enough. But f coincides with g_n on dom g_n, so g’≥f entails that g’≥g_n: for every m in dom g’ ⊆ dom g_n, g’(m)≥f(m)=g_n(m), and for every m outside dom g’, g’(m)=ω. By Fact H, Q_g’ lies above (namely, is included in) Q_gn. But, by (A), Q_gn is in U. Since U is upwards-closed, Q_g’ (= Q’) is in U. Since Q’ is arbitrary in ☐V, this shows that ☐V ⊆ U. ☐

Xiaoyong Xi calls conformal any well-filtered topological space X such that the upper Vietoris and Scott topologies of ⊇ coincide on QX. (I am not sure where this comes from. He gave me some names for the inventors of the notion, but I was unable to find a corresponding reference.)

One can read Lemma 8.3.26 in the book as saying that every well-filtered, locally compact space is conformal. A theorem due to Xiaoquan Xu and Zhongqiang Yang [9, Corollary 5.9] states that every well-filtered, first-countable space such that every compact saturated subset has countably many minimal elements is conformal. Here L_fan is neither core-compact nor first-countable, but is conformal nonetheless. (Note that its compact saturated subsets not only have countably many, but even finitely many minimal elements.)

I will conclude by the following summary: L_fan is an example of a conformal space in its Scott topology, which is not first-countable and not core-compact, but is a consonant, maximal limit space and a complete lattice.

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— Jean Goubault-Larrecq (June 20th, 2024)