PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 64 (78), 1998, 146–152

CAUCHY NETS AND OPEN COLORINGS

Ilijas Farah

Communicated by Žarko Mijajlović

Abstract. The Open Coloring Axiom, OCA, (see [6]) is used to prove that ℝ^{ℕ} equipped with a natural uniform structure is complete, answering a question from [3].

1. Cauchy nets

The Open Coloring Axiom, OCA, was introduced by Todorcevic in [6] in the course of studying the Hausdorff completeness (i.e., gap spectrum) of the structure ℕ^{ℕ} of all integer-valued sequences (see [6, Theorem 8.6], also [2, Chapter IV]. In this note we use the same axiom to deduce another, rather different, completeness property of ℕ^{ℕ}. To express the new kind of completeness, we need to define an abstract distance on the set ℝ^{ℕ} of all sequences of real numbers. First let

ℕ^{↑ℕ} = {g : ℕ → ℕ : lim_{i→∞} g(i) = ∞}.

For g ∈ ℕ^{↑ℕ} let

U_{g} = {⟨x, y⟩ in ℝ^{ℕ} : |x − y| < g + M for some fixed M ∈ ℕ}.

The symbol |x − y| < g + M is interpreted as pointwise inequality, namely as

|x(n) − y(n)| < g(n) + M for all n.

If x, y are such that |x − y| is uniformly bounded by some fixed M ∈ ℕ, we write x ∼ y, and say that in this case x and y are equivalent. The following simple fact explains this terminology.

Lemma 1.1. ⟨x, y⟩ ∈ U_{g} for all g ∈ ℕ^{↑ℕ} if and only if x ∼ y. □

AMS Subject Classification (1991): 54E15, 54A20, 03E35

Therefore, U = {A ⊆ (ℝ^{ℕ})^{2} : A ⊇ U_{g} for some g ∈ ℕ^{↑ℕ}} is a pseudouniformity on ℝ^{ℕ} (see [Kel]), which can naturally be identified with a uniformity on the quotient ℝ^{ℕ}/ ∼. Let us note that ℕ^{ℕ} intersects every ∼-equivalence class, and therefore U can be consider as a pseudouniformity on this space. One can think of the elements of index-set ℕ^{↑ℕ} as different orders of the infinity: Note that the set U_{g} gets finer (i.e. smaller) as the growth rate of g gets slower, and therefore only the slow-growing functions from ℕ^{↑ℕ} matter in the study of U. Since this is exactly the opposite from the way the structure ⟨ℕ^{↑ℕ}, <^{∗}⟩ is usually considered (see e.g. [6]), let us define a transformation Φ: ℕ^{↑ℕ} → ℕ^{↑ℕ} by

Φ(f)(n) = max{k : f(k) ≤ n}.

Recall that for x, y ∈ ℝ^{ℕ} and m ∈ ℕ we write 1. x <^{m} y if x(n) < y(n) for all n ≥ m.

2. x <^{∗} y if x <^{m} y for some m ∈ ℕ.

Then f <^{∗} g implies that Φ(f) >^{∗} Φ(g) and for every h ∈ ℕ^{↑ℕ} there is some g ∈ ℕ^{↑ℕ} such that Φ(g) <^{∗} h. This fact shows that the slow-growing functions behave the same way as the well-studied fast growing functions, and it will be used later (see Claim 2.3.)

A Cauchy net in ⟨ℝ^{ℕ}, U⟩ is a sequence of the form ⟨x_{a} : a ∈ D⟩ for some directed set ⟨D, <⟩ such that for every g ∈ ℕ^{↑ℕ} there is d(g) ∈ D such that

⟨x_{a}, x_{b}⟩ ∈ U_{g}

for all a, b ≥ d(g) in D. This net converges to some x_{∞} ∈ ℝ^{ℕ} if for every g ∈ ℕ^{↑ℕ} there is a ∈ D such that ⟨x_{∞}, x_{b}⟩ ∈ U_{g} for all b > a (see [Kel]). We say that a uniform space ⟨ℝ^{ℕ}, U⟩ is complete if every Cauchy net in this space converges.

The natural question whether the space ⟨ℝ^{ℕ}, U⟩ is complete was raised by Kaufhold in [3], and Watson [8] proved that under the Continuum Hypothesis the answer is negative by constructing an ⟨ω_{1}, ω_{1}⟩-gap one of whose sides is a (necessarily divergent) Cauchy net. Then Steprans [5] proved that Proper Forcing Axiom, PFA, implies that every Cauchy net in ⟨ℝ^{ℕ}, U⟩ converges. The purpose of this note is to prove that the Open Coloring Axiom, OCA, gives the same conclusion. Since OCA is a consequence of PFA, this strengthens Steprans’ result and also shows that the statement “Every Cauchy net in ⟨ℝ^{ℕ}, U⟩ converges” does not have any large cardinal strength.

I would like to thank Stevo Todorčević for several remarks which have considerably improved this note.

2. Cauchy nets under OCA

Let us first recall the statement of OCA (see [6, §8].

OCA: If X is a separable metric space and [X]^{2} = K_{0} ∪ K_{1} is a partition such that K_{0} is an open subset of [X]^{2} = {{x, y} : x ≠ y, x, y ∈ X} then one of the following applies:

(a) There is an uncountable Y ⊆ X which is K_{0}-homogeneous, i.e., [Y]^{2} ⊆ K_{0} (b) X can be covered by countably many sets, each of which is K_{1}^{n}-homogeneous

(X is σ-K_{1}-homogeneous).

Our proof of the following theorem should be compared with the proof of [6, Theorem 8.7].

Theorem 2.1. OCA implies that the uniform space ⟨ℝ^{ℕ}, U⟩ is complete.

Proof. We need to prove that every Cauchy net in ⟨ℝ^{ℕ}, U⟩ converges. Let us start by giving an alternative definition of the pseudouniformity U. For g ∈ ℕ^{↑ℕ} define a subset of ℝ^{ℕ} × ℝ^{ℕ} as follows:

U_{g}^{∗} = {⟨x, y⟩ : |x − y| <^{∗} g},

and consider the pseudouniformity U^{∗} = {A ⊆ (ℝ^{ℕ})^{2} : A ⊇ U_{g}^{∗} for some g ∈ ℕ^{↑ℕ}}.

Lemma 2.2. The uniformities U and U^{∗} coincide.

Proof. It suffices to note that

U_{g/2} ⊆ U_{g}^{∗} ⊆ U_{g}.

To see the left-hand side inequality, note that |x−y| < g/2+M implies |x−y| <^{k} g, where k is large enough so that g(k) > 2M. For the right-hand side inequality, observe that |x − y| <^{k} g implies |x − y| < g + max{g(i) : i ≤ k}. □

Fix a Cauchy net ⟨x_{a} : a ∈ D⟩. For g ∈ ℕ^{↑ℕ} pick d(g) ∈ D such that

|x_{a} − x_{b}| <^{∗} g

for a, b ≥ d(g). For x ∈ ℝ^{ℕ} and g ∈ ℕ^{↑ℕ} let

B_{g}(x) = {y : |x − y| < g}

Define a partition [ℕ^{↑ℕ}]^{2} = K_{0} ∪ K_{1} by letting {f, g} ∈ K_{0} if

x_{d(f)}(m) − x_{d(g)}(m) > 2f(m) + 2g(m) + 2

for some m ∈ ℕ. (Or equivalently, \{f, g\} ∈ K_{1} if x_{d(g)} ∈ \overline{B_{2f+2g+2}(x_{d(f)})}.) If we consider ℕ^{↑ℕ} as a subspace of ℕ^{↑ℕ} × ℝ^{ℕ} by identifying g ∈ ℕ^{↑ℕ} with the pair ⟨g, x_{d(g)}⟩, then K_{0} becomes an open subset of [ℕ^{↑ℕ}]^{2}.

Assume first that ℕ^{↑ℕ} can be covered by countably many K_{1}-homogeneous subsets. Since the poset ⟨ℕ^{↑ℕ}, >^{∗}⟩ is σ-directed, one of these subsets, call it H, is cofinal in this poset (namely, for every g ∈ ℕ^{↑ℕ} there is g′ ∈ H such that g >^{∗} g′). Consider the intersection

⋂_{g∈H} \overline{B_{2g+2}(x_{d(g)})}.

We claim that this set is nonempty. To see this, it will suffice to prove that for every m the intersection of family of intervals

I_{g,m} = [x_{d(g)}(m) − 2g(m) − 2, x_{d(g)}(m) + 2g(m) + 2] (g ∈ H),

is nonempty. But by the K_{1}-homogeneity we have

|x_{d(f)}(m) − x_{d(g)}(m)| ≤ 2f(m) + 2g(m) + 4,

and therefore every two intervals from the family intersect, and by Helly’s theorem every finite subfamily of this family has the nonempty intersection. By compactness, the intersection of the whole family, ⋂_{g∈h} I_{g,m}, is nonempty for every m, and therefore we can pick

x_{∞} ∈ ∏_{m=1}^{∞} ⋂_{g∈H} I_{g,m} = ⋂_{g∈h} \overline{B_{2g+2}(x_{d(g)})}.

Towards proving that our Cauchy net converges to x_{∞}, fix g ∈ ℕ^{↑ℕ} and find g′ <^{∗} g/2 − 2 in H. Then for every a ≥ d(g′) we have

|x_{∞} − x_{a}| < |x_{∞} − x_{d(g′)}| + |x_{d(g′)} − x_{a}| <^{∗} 2g′ + 2 <^{∗} g.

Now assume ℕ^{↑ℕ} can not be covered by countably many K_{1}-homogeneous sets. Since K_{0} is open, OCA implies that ℕ^{↑ℕ} includes an uncountable K_{0}-homo-geneous set, g_{ξ} (ξ < ω_{1}).

Claim 2.3. There is a g_{ω_{1}} in ℕ^{↑ℕ} such that g_{ω_{1}} <^{∗} g_{ξ} for all ξ.

Proof . Let Φ: ℕ^{↑ℕ} → ℕ^{↑ℕ} be the function defined in the introduction. Pick f_{ξ} such that Φ(f_{ξ}) <^{∗} g_{ξ}. Recall that OCA implies that there is an f_{ω_{1}} >^{∗} f_{ξ} for all ξ (see [6, Theorem 8.7]. Then g_{ω_{1}} = Φ(f_{ω_{1}}) is as required. □

Consider functions x_{ξ} = x_{d(g_{ξ})} (ξ ≤ ω_{1}). Since D is directed, for every ξ there is a_{ξ} > d(g_{ξ}), d(g_{ω_{1}}), and by the definition of d(·) we have

|x_{ξ} − x_{ω_{1}}| < |x_{ξ} − x_{a_{ξ}}| + |x_{a_{ξ}} − x_{ω_{1}}| <^{∗} g_{ξ} + g_{ω_{1}} <^{∗} 2g_{ξ}.

Let n_{ξ} ∈ ℕ be such that |x_{ξ} − x_{ω_{1}}| <^{n_{ξ}} 2g_{ξ}. By going to an uncountable subset, we can assume that n_{ξ} = n̄ and that ⌈x_{ξ}(i)⌉ = ⌈x_{η}(i)⌉ = s̄_{ξ}(i) for all i ≤ n̄. for all ξ and some fixed n̄ ∈ ℕ and s̄: {1,…, n̄} → ℕ. Therefore if m > n̄ we have

|x_{ξ}(m) − x_{η}(m)| < 2g_{ξ}(m) + 2g_{η}(m)

for all ξ, η < ω_{1}. Moreover, for m ≤ n̄ we have |x_{ξ}(m)−x_{η}(m)| < 2 for all ξ, η < ω_{1}, and therefore |x_{ξ} − x_{η}| < 2g_{ξ} + 2g_{η} + 2, or equivalently, {g_{ξ}, g_{η}} ∈ K_{1}. But this is a contradiction, and it completes the proof. □

3. OCA_{∞}

Our first proof of Theorem 2.1 used a strengthening of OCA which was extracted from Steprans’ proof [5]. Although it turned out that Theorem 2.1 already follows from Todorčevic’s OCA, this axiom may turn out to be interesting in its own right.

If X is a topological space, then

[X]^{2} = K_{0}^{n} ∪ K_{1}^{n}, n ∈ ℕ

is a decreasing sequence of open partitions if every K_{0}^{n} is an open subset of [X]^{2} (in a natural topology induced by the topology on X) and K_{0}^{n} ⊇ K_{0}^{n+1} for all n.

OCA_{∞}: If X is a separable metric space and [X]^{2} = K_{0}^{n} ∪ K_{1}^{n}, n ∈ ℕ, is a decreasing sequence of open partitions then one of the following applies:

(a) X = ⋃_{n∈ℕ} F_{n}, where each F_{n} is K_{1}^{n}-homogeneous (in this case we say that X is σ-K_{1}^{∗}-homogeneous).

(b) There is an uncountable Y ⊆ X which is covered by countably many K_{0}^{n}- homogeneous sets (Y is σ-K_{0}^{n}-homogeneous) for every n. Note that the requirement that sequence K_{0}^{n} is decreasing is, in some sense,

necessary, for if ⋂_{i∈s} K_{0}^{i} = ∅ for some finite set s then clearly no uncountable set can be simultaneously K_{0}^{i}-homogeneous for all i ∈ F.

We do not know whether OCA_{∞} (or its stronger version, see (b′) below) follows from OCA. We can consider a strengthening of OCA_{∞}, obtained when (b) is replaced by: (b′) There is an uncountable Z ⊆ {0, 1}^{ℕ} and y_{α} ∈ X (α ∈ Z) such that (recall

that Δ(α, β) is the minimal n such that α(n) ≠ β(n)):

{y_{α}, y_{β}} ∈ K_{0}^{Δ(α,β)}

for all α, β ∈ Z.

If (b′) is satisfied, then set Y = {y_{α} : α ∈ Z} satisfies a strong form of (b), because for every s ∈ {0, 1}^{n} the set

Y_{s} = {y_{α} : α ∈ [s]}

is K_{0}^{n}-homogeneous, and therefore Y is covered by 2^{n} many K_{0}^{n}-homogeneous sets for every n.

The consistency of OCA_{∞}, or its stronger version defined above, can be proved in the same way as the consistency of OCA (see [6]), by using the following lemma instead of [6, Theorem 4.4].

Lemma 3.1. Assume CH, let X be a separable metric space and let [X]^{2} = K_{0}^{n}∪K_{1}^{n} be a decreasing sequence of open partitions. If X is not σ-K_{1}^{∗}-homogeneous, then there is a ccc poset P = P(X, K_{0}^{∗}) which forces (b′)

Proof. By using a diagonalization argument of [6, Theorem 4.4], we can find an uncountable subset Y of X such that poset of finite K_{0}^{n}-homogeneous subsets of Y is powerfully ccc (i.e. every its finite power is ccc) for every n. Define poset P so that its typical condition is p = ⟨F^{p}, g^{p}⟩ = ⟨F, s⟩, where

1. F is a finite subtree of {0, 1}^{<ℕ},

2. g is an injection of top-nodes of F into Y

3. {g(s), g(t)} ∈ K_{0}^{Δ(s,t)} for every pair s, t of distinct top-nodes of F. The ordering on P is defined by letting ⟨F^{p}, g^{p}⟩ ≤ ⟨F^{q}, g^{q}⟩ if

4. F^{p} ⊇ F^{q} and g^{p}(t) = g^{q}(s) whenever s ⊆ t.

To see that P is ccc, fix ⟨F^{ξ}, g^{ξ}⟩ (ξ < ω_{1}) in P. We can assume that F^{ξ} = F̄ for some fixed tree F̄ and all ξ. Let n̄ be the height of F̄ and let t_{1},…, tk̄ be its top-nodes. Finite sets {g^{ξ}(t_{1}),…, g^{ξ}(tk̄)} can be considered as conditions in k-th power of a poset of finite Kn̄ ; 0-homogeneous subsets of Y. Since this poset is powerfully ccc, we can find ξ < η such that

{g^{ξ}(t_{i}), g^{η}(t_{i})} ∈ Kn̄ ; 0

for all i = 1,…, k̄. Now define an extension ⟨F, g⟩ of ⟨F̄, g^{ξ}⟩ and ⟨F̄, g^{η}⟩ as follows: Let F be the end-extension of F̄ obtained by adding nodes t_{i}^{ξ} and t_{i}^{η} above each t_{i}, so that t_{i}^{j} (i ≤ k, j = ξ, η) are the top-nodes of F. Define g by letting

g(t_{i}^{ζ}) = g^{ζ}(t_{i}), for ζ = ξ, η and i = 1,… k̄.

The choice of n̄ assures that Δ(t_{i}^{ξ}, t_{i}^{η}) ≥ n̄ and therefore condition 3. is satisfied. Therefore ⟨F, g⟩ extends both ⟨F^{ξ}, g^{ξ}⟩ and ⟨F^{η}, g^{η}⟩, and poset P is ccc.

If G is a P-generic filter, let Y_{G} = {g^{p}(t) : p ∈ G}, and for y ∈ Y_{G} let

α_{G}(y) = ⋃{t ∈ {0, 1}^{<ℕ} : g^{p}(t) = y for some p ∈ G}.

Note that, by 4., the set on the right-hand side will be a branch (infinite, by genericity) of {0, 1}^{<ℕ}, and therefore Z_{G} = {α_{G}(y) : y ∈ Y_{G}} is a subset of a Cantor space. If p is a condition in P which forces Y to be uncountable; then p forces that Z_{G} and Y_{G} satisfy (b′). □

Remark 3.2. One of the reasons why can OCA be considered as a natural axiom is the fact that it has a definable version (see [TF], [Fe]). Let us note that a definable version of OCA_{∞} is also true. Namely, if A is an analytic subset of some Polish space and {K_{0}^{n}} is a decreasing sequence of open partitions of [A]^{2}, then either A is σ-K_{1}^{∗}-homogeneous or there is a continuous embedding g of a Cantor space into A such that {g(α), g(β)} ∈ K_{0}^{Δ(α,β)} for all distinct α, β.

References

1. I. Farah, Analytic quotients, 1997, submitted

2. Q. Feng, Homogeneity for open partitions of reals, Trans. Amer. Math. Soc. 339 (1994), 659–684.

3. T. Kaufhold, Ph. D. Thesis, University of Jena, 1997

4. J.L. Kelley, General Topology, Van Nostrand, 1955

5. J. Steprans, Unpublished manuscript, 1997

6. S. Todorčevic, Partition Problems in Topology, AMS, Providence, Rhode Island, 1989

7. S. Todorcevic and I. Farah, Some Applications of the Method of Forcing, Mathematical Institute, Belgrade and Yenisei, Moscow, 1995

8. S. Watson, Unpublished manuscript, 1997

Department of Mathematics (Received 15 04 1998) York University (Revised 15 09 1998) North York Canada M3J 1P3

ifarah@mathstat.yorku.ca

and

Matematički institut Kneza Mihaila 35 11001 Beograd, p.p. 367 Yugoslavia