A NEW SUBCONTINUUM OF βℝ \ ℝ

ALAN DOW AND KLAAS PIETER HART

Abstract. We present a method for describing subcontinua of βℝ \ ℝ. This method enables us to construct in ZFC a new subcontinuum of βℝ \ ℝ.

We also show that the nontrivial layers of standard subcontinua can be described by our method. This allows us to construct a layer with a proper dense F_{σ}-subset and bring the number of (known) nonhomeomorphic subcontinua of βℝ \ ℝ to 14.

Introduction

It is fairly well-known that the Čech-Stone remainder ℍ^{∗} of the half line ℍ = [0, ∞) is an indecomposable continuum — a fact that was proved by Woods in [1968] and Bellamy in [1971]. Much of the structure of ℍ^{∗} is known but many questions remain.

In this paper we continue the investigation into the number of topologically different subcontinua of ℍ^{∗}. So far through the efforts of Van Douwen [1977], Smith [1986] and Zhu [1993] nine different subcontinua of ℍ^{∗} have been discovered, in ZFC. We shall describe these continua briefly in Subsection 1.6.

The purpose of this paper is to construct, in ZFC, a new (tenth) subcontinuum of ℍ^{∗} and show how to use this continuum to raise, again in ZFC, the number of different subcontinua of ℍ^{∗} to fourteen. We do this by showing how, under CH, our continuum can be seen as a layer in a standard subcontinuum; because we have essentially four different layers in standard subcontinua (so far) this gives us four extra decomposable subcontinua. Under ¬CH we appeal to Dow [1984] for at least six more decomposable subcontinua.

As a prelibation of things to come we give an outline of the rest of the paper. In Section 2 we show how, with the aid of certain subsets of ω^{∗}, one can parameterize the indecomposable subcontinua of ℍ^{∗}. In Section 3 we use this parameterization with a minimal closed and σ-invariant subset of ω^{∗} as input to construct our new continuum. Here σ is the selfmap of ω^{∗} determined by the shift on ω. We also show that our continuum has 2^{𝔠} composants, thus providing a ‘naturally occurring’ continuum of this type. In Section 4 we prove in ZFC that ℍ^{∗} has at least fourteen topologically different subcontinua although the families differ according to whether CH holds or not. Finally, in Section 5 we make some remarks and pose some questions.

1980 Mathematics Subject Classification (1985 Revision). Primary Primary: 54D40, 54F15.

Secondary: 04A30, 54G05.

Key words and phrases. continuum, indecomposable continuum, shift on ω^{∗}, Continuum Hypothesis.

1. Preliminaria

In this section we establish some notation and recall some facts that will be needed later on. We refer to Hart [1992] for proofs and proper references.

1.1. The shift on ω^{∗}. The shift σ on ω^{∗} is the map determined by the usual shift on ω: σ(n) = n + 1. For a point u of ω^{∗} its image under σ is generated by the family {A + 1 : A ∈ u}; because of this we sometimes write u + 1 for σ(u). Likewise we write u − 1 for σ^{←}(u).

We will adopt the slightly nonstandard convention of calling a subset F of ω^{∗} σ-invariant if σ[F] ⊆ F and σ^{←}[F] ⊆ F (nonstandard because normally one takes σ-invariant to mean that σ[F] ⊆ F only). The reason for this will become clear in Section 2; the closed and σ-invariant subsets occur naturally when we parameterize indecomposable subcontinua of ℍ^{∗}.

We note that, with our interpretation, the set of accumulation points of a forward orbit is always σ-invariant. For let q be an accumulation point of cl{p + n : n ∈ ω}; there is an ultrafilter u ∈ ω^{∗} such that q = u-lim(p + n). But then q − 1 = u-lim(p + (n − 1)) and q + 1 = u-lim(p + (n + 1)). Here, for a sequence ⟨x_{n}⟩_{n} and u ∈ ω^{∗}, we say x = u-lim x_{n} iff for every neighbourhood U of x the set {n : x_{n} ∈ U} belongs to u.

1.2. The space 𝕄. The remainder 𝕄^{∗} of the space 𝕄 = ω × 𝕀 determines the structure of ℍ^{∗} completely. Before we show this we fix some notation. We abbreviate {n} × 𝕀 by 𝕀_{n} and we let π : β𝕄 → βω be the map determined by π(n, x) = n. For u ∈ ω^{∗} the fiber π^{←}(u) is denoted 𝕀_{u} (it is in a natural sense the u-th term of the sequence ⟨𝕀_{n} : n ∈ ω⟩: in the hyperspace of closed subsets of β𝕄 one has 𝕀_{u} = u-lim 𝕀_{n}). Similarly, if x ∈ ^{ω}𝕀 then we use x_{u} to denote the point u-lim⟨n, x(n)⟩ of 𝕀_{u}.

The set 𝕀_{u} is a continuum that is irreducible between the two points 0_{u} = u-lim⟨n, 0⟩ and 1_{u} = u-lim⟨n, 1⟩. That 𝕀_{u} is irreducible between 0_{u} and 1_{u} follows from the fact that {x : x ∈ ^{ω}𝕀} is a dense set of cut points of 𝕀_{u}. There is a natural quasi-order on 𝕀_{u}: one says x ⩽_{u} y if every subcontinuum of 𝕀_{u} that contains 0_{u} and y also contains x. We shall employ the usual interval notation for this ordering. We note that if x, y ∈ ^{ω}𝕀 then x_{u} ⩽_{u} y_{u} iff {n : x(n) ⩽ y(n)} ∈ u.

This ordering ⩽_{u} divides 𝕀_{u} into layers: the equivalence classes under the relation

x ≡ y  iff  x ⩽_{u} y and y ⩽_{u} x.

With a layer L one associates two subsets of ^{ω}𝕀:

A_{L} = {a ∈ ^{ω}𝕀 : L ∩ cl[0, a] = ∅}

and

B_{L} = {b ∈ ^{ω}𝕀 : L ∩ cl[b, 1] = ∅}.

Here, for x, y ∈ ^{ω}𝕀, the symbol [x, y] abbreviates ⋃_{n}{n}×[x(n), y(n)]; observe that [x_{u}, y_{u}] = cl[x, y] ∩ 𝕀_{u}.

The pair ⟨A_{L}, B_{L}⟩ represents a gap in the linearly ordered set ^{ω}𝕀/u. The gap is filled iff L = {x_{u}} for some x ∈ ^{ω}𝕀. If the gap is unfilled then one can show that L consists of more than one point iff there is a function f : ω → ℕ such that {n : b(n) − a(n) > 1/f(n)} belongs to u whenever a ∈ A_{L} and b ∈ B_{L}.

If L is such that [0_{u}, L) has countable cofinality then L does consist of more than one point. For take a cofinal sequence ⟨a_{n}⟩_{n} in [0_{u}, L); its set of accumulation points D is contained in L and, because 𝕄^{∗} is an F -space, homeomorphic to ω^{∗}. We also note that in this case the interval (L, 1_{u}] necessarily has uncountable coinitiality. In other words, the gap ⟨A_{L}, B_{L}⟩ is not of type ⟨ω, ω⟩.

1.3. Parameterizing ℍ^{∗}. The map q : β𝕄 → βℍ determined by q(n, x) = n + x maps 𝕄^{∗} onto ℍ^{∗}. Because q is one-to-one on every 𝕀_{n} it is also one-to-one on every 𝕀_{u} and because it only identifies (1, n) and (0, n + 1) for every n it also only identifies the points 1_{u} and 0_{u+1} for every u ∈ ω^{∗}. One can readily check these statements using standard properties of the Čech-Stone compactification.

One can define a map like q using any sequence a that increases to infinity: Simply put q_{a}(n, x) = a_{n} + x(a_{n+1} − a_{n}); we shall call q_{a} the parameterization of ℍ^{∗} determined by a. The map q from the previous paragraph will be called the standard parameterization.

1.4. Standard subcontinua. If q_{a} : 𝕄^{∗} → ℍ^{∗} is a parameterization then, as noted before, the map q_{a} is one-to-one on every continuum 𝕀_{u}. Therefore q_{a}[𝕀_{u}] and 𝕀_{u} are homeomorphic; we call a subcontinuum of ℍ^{∗} of the form q_{a}[𝕀_{u}] a standard subcontinuum of ℍ^{∗}.

The basic facts about standard subcontinua that we shall use here are that every proper subcontinuum K of ℍ^{∗} is the intersection of the family of all standard subcontinua that contain it and if K is nontrivial then it contains at least one standard subcontinuum. Theorem 2.6 substantially improves this last sentence.

Usually standard subcontinua are defined via embeddings of β𝕄 into βℍ but one readily checks that the two approaches are equivalent.

1.5. Decomposable versus indecomposable. A continuum is decomposable if it can be written as the union of two proper subcontinua and indecomposable otherwise. One can see whether a subcontinuum of ℍ^{∗} is decomposable or not by looking at its position inside standard subcontinua.

Lemma 1.1. A subcontinuum of ℍ^{∗} is decomposable iff it is a nondegenerate interval of some standard subcontinuum.

Here we take the intervals with respect to the quasi-order introduced before. It follows that decomposable subcontinua have cut points: intervals in standard subcontinua have (relative) interior and hence contain one of the cut points from Subsection 1.2. So, whenever an indecomposable continuum sits inside a standard subcontinuum it sits inside a layer of it. In fact layers of standard subcontinua are themselves indecomposable. Lemma 1.1 may also be used to prove the following one.

Lemma 1.2. If of the two subcontinua K and L one is indecomposable and if their intersection is nonempty then either K ⊆ L or L ⊆ K.

We shall use Lemma 1.2 frequently in showing that one continuum must be contained in another one.

1.6. The known subcontinua. As noted in the introduction up to now nine nonhomeomorphic subcontinua of ℍ^{∗} were known in ZFC. Because we will want to show that the continuum that we will construct in Section 3 is new we give a brief description of the nine continua.

The first, K_{1}, consists of one point.

The next six will be constructed from a standard subcontinuum. Let us fix one such continuum 𝕀_{u}. We take K_{2} = 𝕀_{u}. Next we take a layer L_{1} such that the interval [0_{u}, L_{1}) has countable cofinality and we let K_{3} = [0_{u}, L_{1}] and K_{4} = [L_{1}, 1_{u}]. Next we take L_{2} <_{u} L_{1} such that (L_{2}, 1_{u}] has countable coinitiality and we let K_{5} = [L_{2}, L_{1}]. Finally we take L_{3} >_{u} L_{1} such that (L_{3}, 1_{u}] has countable coinitiality as well and we let K_{6} = [L_{2}, L_{3}] and K_{7} = [L_{1}, L_{3}]. These six continua are all decomposable and have two distinguished end sets; the number of one-point-or G_{δ}-end sets may be used to show that they are mutually nonhomeomorphic; for example K_{3} and K_{4} are nonhomeomorphic: K_{3} has a one-point end set and a G_{δ}- end set, namely L_{1}, but in K_{4} the end set L_{1} is not a G_{δ}-set nor does it consist of one point.

The continuum K_{8} will be L_{1} and the ninth continuum K_{9} is such that it has an increasing sequence ⟨C_{n}⟩_{n} of proper indecomposable subcontinua such that K_{9} = cl ⋃_{n} C_{n}. These two continua are indecomposable and distinguished by the presence of a proper dense F_{σ}-subset — K_{9} has one and K_{8} has none.

2. Making subcontinua from shift-invariant subsets of ω^{∗}

In this section we describe a method by which all indecomposable subcontinua of ℍ^{∗} may be constructed. The idea is quite simple: For a subset F of ω^{∗} we abbreviate ⋃_{u∈F} 𝕀_{u} by 𝕄_{F} and we will denote the image of 𝕄_{F} under q by C_{F}. We shall identify F with the set {0_{u} : u ∈ F } and also with the image of this set under q.

We shall see that C_{F} is an indecomposable continuum whenever F is closed, σinvariant, dense-in-itself and not the union of two proper closed disjoint σ-invariant subsets. Conversely, if K is an indecomposable subcontinuum of ℍ^{∗} then there are a subset F of ω^{∗} and a piecewise linear autohomeomorphism h of ℍ such that K = h[C_{F}].

The first lemma enables us to recognize σ-invariant subsets of ω^{∗} by the behavior of 𝕄_{F} and C_{F}.

Lemma 2.1. The set F is σ-invariant iff 𝕄_{F} = q^{←}[C_{F}].

Proof. Assume F is σ-invariant and let x ∈ q^{←}[C_{F}]. Choose y ∈ 𝕄_{F} and u ∈ F such that q(y) = q(x) and y ∈ 𝕀_{u}. If y ≠ x then either y = 0_{u} and x = 1_{u−1} or y = 1_{u} and x = 0_{u+1}; in either case x ∈ 𝕄_{F} because u − 1, u + 1 ∈ F.

Conversely assume 𝕄_{F} = q^{←}[C_{F}] and let u ∈ F. Because q(1_{u}) = q(0_{u+1}) we must have 0_{u+1} ∈ 𝕄_{F} but this means that u + 1 ∈ F. Likewise one concludes that u − 1 ∈ F.

Now we can show prove the following proposition.

Proposition 2.2. If C_{F} is an indecomposable continuum then F is closed and σ-invariant.

Proof. Let u ∈ F. Because q[𝕀_{u+1}] meets q[𝕀_{u}] it also meets C_{F}. It can not contain C_{F} because it does not contain q[𝕀_{u}], therefore q[𝕀_{u+1}] ⊆ C_{F} and so u + 1 ∈ F. Likewise one shows that u − 1 ∈ F.

Finally, F is closed because F = π[q^{←}[C_{F}]] and π is a closed map.

Next we need to know for which closed σ-invariant subsets F of ω^{∗} the space C_{F} is connected.

Theorem 2.3. The set C_{F} is connected iff F can not be written as the union of two disjoint proper closed σ-invariant subsets.

Proof. Necessity: If F = G ⊕ H with G and H both σ-invariant then 𝕄_{F} = 𝕄_{G} ⊕ 𝕄_{H} hence C_{F} = C_{G} ⊕ C_{H} by Lemma 2.1.

Sufficiency: If C_{F} = G⊕H the we can divide F into two sets G′ = {u : q[𝕀_{u}] ⊆ G} and H′ = {u : q[𝕀_{u}] ⊆ H}. Because every 𝕀_{u} is connected this is a partition of F.

For every u ∈ F the set q[𝕀_{u−1}] ∪ q[𝕀_{u}] ∪ q[𝕀_{u+1}] is connected, so if u is in G′ (or H′) then so are u − 1 and u + 1. This shows that G′ and H′ are σ-invariant.

To see that G′ and H′ are closed just observe that G = C_{G′} and so (again by the Lemma) G′ = π[q^{←}[G]] and likewise for H.

We see that every ‘indecomposable’ σ-invariant closed set determines a subcontinuum of ℍ^{∗}. Next we determine when C_{F} is an indecomposable continuum.

Lemma 2.4. If u ∈ F then u is an isolated point of F iff q[𝕀_{u}] has nonempty interior in C_{F}.

Proof. If u is isolated in F then 𝕀_{u} \ {0_{u}, 1_{u}} is open in 𝕄_{F} and saturated with respect to q so its image is open if C_{F} and contained in q[𝕀_{u}].

Conversely, if there is an open set O such that ∅ ≠ O ∩ C_{F} ⊆ q[𝕀_{u}] then q^{←}[O] does the same thing for 𝕀_{u} with respect to 𝕄_{F}. Pick an open rectangle O(a, b, U) in 𝕄^{∗} such that O(a, b, U) ∩ 𝕄_{F} is nonempty and contained in 𝕀_{u}. Then U ∩ F = {u}.

Now we can characterize indecomposability of C_{F}.

Theorem 2.5. If C_{F} is a continuum then it is indecomposable iff F is dense-in-itself.

Proof. We just proved necessity: If u ∈ F is isolated then 𝕀_{u} is a subcontinuum of C_{F} with nonempty interior (if F = {u} then certainly C_{F} = 𝕀_{u} is decomposable).

To prove sufficiency assume that C_{F} is decomposable. By Lemma 1.1 we can find a standard subcontinuum 𝕁_{u} such that C_{F} has nonempty interior in 𝕁_{u} and therefore contains a cut point x of 𝕁_{u}. Choose v ∈ F such that 𝕀_{v} contains x. Then 𝕀_{v} is a subinterval of 𝕁_{u} and hence has nonempty interior in 𝕁_{u}. So certainly 𝕀_{v} has nonempty interior in C_{F}; it follows that v is an isolated point of F.

We conclude that we get an indecomposable continuum in ℍ^{∗} each time we get a dense-in-itself indecomposable σ-invariant subset of ω^{∗}. The converse holds as well.

Theorem 2.6. Let K be a nontrivial indecomposable subcontinuum of ℍ^{∗}. Then there are a σ-invariant subset F of ω^{∗} and a (piecewise linear) autohomeomorphism h of ℍ such that K = βh[C_{F}].

Proof. Take a standard subcontinuum that is contained in K, that is take a parameterization q_{a} of ℍ and u ∈ ω^{∗} such that q_{a}[𝕀_{u}] ⊆ K.

We claim that for every point v ∈ ω^{∗} either q_{a}[𝕀_{v}] ⊆ K or q_{a}[𝕀_{v}] ∩ K = ∅. For if q_{a}[𝕀_{v}] meets K then it can not contain K as it can not contain q_{a}[𝕀_{u}], so it must be contained in K.

Now we consider F = {v ∈ ω^{∗} : q_{a}[𝕀_{v}] ⊆ K}. From what we have established above it follows that K = q_{a}[𝕄_{F}] = βh[C_{F}], where h : ℍ → ℍ is the unique piecewise linear map for which h ∘ q = q_{a}.

Because C_{F} apparently is an indecomposable continuum we see that F must be σ-invariant.

Remark 2.7. Let us note that we cannot distinguish nontrivial the subcontinua of ℍ^{∗} by means of cellularity, density or (π-)weight.

As ℍ^{∗} has weight 𝔠 it suffices to show that each nontrivial subcontinuum has cellularity 𝔠. Indeed, for every x ∈ (0, 1) we can take the open subset O_{x} of 𝕄^{∗} determined by ⋃_{n}{n} × (x − 2^{−n}, x + 2^{−n}); the family {O_{x} : x ∈ (0, 1)} is pairwise disjoint and every O_{x} meets every 𝕀_{u}.

Now if K is a subcontinuum of ℍ^{∗} then we take a parameterization q_{a} : 𝕄 → ℍ and u ∈ ω^{∗} such that q_{a}[𝕀_{u}] ⊆ K. Then {q[O_{x}] ∩ K : x ∈ (0, 1)} is a family of pairwise disjoint nonempty open subsets of K. (Note that O_{x} = q\overleftarrow{a}[q_{a}[O_{x}]] for every x, so that q_{a}[O_{x}] is open in ℍ^{∗}.)

3. A New Continuum

In this section we apply the construction from Section 2, with very special input, to construct a new subcontinuum of ℍ^{∗}. The input that we shall use is a minimal closed σ-invariant subset of ω^{∗}. Such sets are easily constructed: Apply Zorn’s Lemma to the family of all closed σ-invariant subsets of ω^{∗}.

It is readily verified that a closed set F is minimally σ-invariant iff for every u ∈ F the orbit O_{u} = {u + n : n ∈ ℤ} is dense in F. From this one easily deduces that for every u ∈ F the set S_{u} = ⋃_{n} q[𝕀_{u+n}] is proper dense F_{σ}-subset of C_{F}.

Each minimal σ-invariant subset is clearly indecomposable and dense-in-itself so it follows that C_{F} is an indecomposable continuum whenever F is closed and minimally σ-invariant. The following theorem shows that C_{F} is indeed new.

Theorem 3.1. If F is a minimal closed σ-invariant subset of ω^{∗} then C_{F} is not homeomorphic to any one of the continua K_{1} through K_{9}.

Proof. Since K_{1} consists of one point and since the continua K_{2}, …, K_{7} are decomposable we only have to deal with K_{8} and K_{9}. Now K_{8} has the property that every nonempty G_{δ}-subset of it has nonempty interior but K_{9} and C_{F} do not have this property, so we are left with K_{9}.

Remember that K_{9} has a proper, dense and meager F_{σ}-subset H = ⋃_{n} C_{n}, where each C_{n} is indecomposable and nowhere dense in C_{n+1}. Assume h : K_{9} → C_{F} is a homeomorphism and choose x ∈ h[C_{0}]; also pick u ∈ F such that x ∈ q[𝕀_{u}].

Because h[H] is dense in C_{F} we know that there is an n such that h[C_{n}] ⊈ q[𝕀_{u}] and consequently — because C_{n} is indecomposable — q[𝕀_{u}] ⊆ h[C_{n}]. But now one shows inductively that q[𝕀_{u+k}] ⊆ h[C_{n}] for every k and hence that S_{u} ⊆ h[C_{n}].

This leads to a contradiction since S_{u} is dense in C_{F} and C_{n} is nowhere dense in K_{9}.

Henceforth we shall denote the continuum of Theorem 3.1 by K_{10}.

The composant of a point x in a continuum X is the union of all proper subcontinua of X that contain the point. In an indecomposable continuum the composants are the equivalence classes under the relation “there is a proper subcontinuum containing both x and y”.

The following proposition identifies the composants of K_{10}. In its proof we apply Theorem 5.3 from Hart [1992] (due to Van Douwen) which gives exact information on how two standard subcontinua intersect (if they intersect). Indeed if K and L are standard subcontinua, K with end points a and b, and L with end points c and d that intersect then K ∩ L equals L if c, d ∈ K, it equals K if c, d ∉ L, and it equals one of the standard subcontinua [a, d], [c, b], [a, c] and [d, b] depending on which of the end points of L is in K and which of the end points of K is in L. Also, if K and L intersect then K ∪ L is a standard subcontinuum as well.

Proposition 3.2. If u ∈ F then S_{u} is the composant of u in K_{10}.

Proof. Denote the composant of u by K_{u}. Clearly S_{u} ⊆ K_{u}, because S_{u} is the union of a family of proper subcontinua of K_{10} that contain the point u.

Conversely, let K be a nontrivial proper subcontinuum of K_{10} that contains u. If K is indecomposable then either K ⊆ q[𝕀_{u}] or q[𝕀_{u}] ⊆ K. The latter case will give rise to a contradiction; for much as in the proof of Proposition 2.2 we can show that for each n ∈ ℤ if q[𝕀_{u+n}] ⊆ K then also q[𝕀_{u+n−1}] ⊆ K and q[𝕀_{u+n+1}] ⊆ K. This would imply that K_{10} = cl S_{u} ⊆ K.

So assume K is decomposable and choose a standard subcontinuum 𝕁_{v} such that K is a nondegenerate interval of it. Now if for all n > 0 we would have u + n ∈ 𝕁_{v} then, by the remarks preceding this proposition we would also have q[𝕀_{u+n}] ⊆ 𝕁_{v} for all n ∈ ω. But then

K_{10} ⊆ cl ⋃_{n⩾0} q[𝕀_{u+n}] ⊆ 𝕁_{v}.

Now K ⊆ K_{10} implies that K_{10} would be a nondegenerate interval of 𝕁_{v} and hence decomposable.

We conclude that there is n > 0 such that u+n ∉ 𝕁_{v} and likewise that u+m ∉ 𝕁_{v} for some m < 0. Applying the remarks preceding the proposition again we see that 𝕁_{v} is contained in the standard subcontinuum ⋃_{m⩽i<n} q[𝕀_{u+i}] and hence that K ⊆ S_{u}.

Thus composants of K_{10} correspond to orbits in F and we see that K_{10} has 2^{𝔠} composants — the first continuum with 2^{𝔠} composants was constructed by Smith in [1976]. This is of interest because the number of composants of ℍ^{∗} and of all countable cofinality layers is independent of ZFC: Under CH the number is 2^{𝔠} and under NCF the number is 1 — we refer to Hart [1992] for references. We do not know the number of composants of K_{9}.

4. Four more subcontinua from CH

In this section we show how our construction may be employed to produce, using CH, four more subcontinua of ℍ^{∗}. At the end of this section we use a result of Dow to show that under ¬CH there are always at least six continua different from K_{1} through K_{10}.

We shall show that, under CH, the continuum K_{10} can be constructed as a layer in some standard subcontinuum 𝕀_{u}. To this end we first investigate how we can make the parameterization of the layers as in Section 2 more concrete.

4.1. Concrete descriptions of layers. We fix a nontrivial layer L of some standard subcontinuum 𝕀_{u} and we find F such that L = C_{F} (more precisely, we find a parameterization q of ℍ and F ⊆ ω^{∗} such that L = q[𝕄_{F}]).

Instead of ω we consider the set K_{f} = {⟨n, m⟩ : m < f(n)}, where f is as in Subsection 1.2, and order it lexicographically. The shift σ on K_{f} is defined by

σ(n, m) = \begin{cases}⟨n, m + 1⟩ & if m + 1 < f(n) and \\ ⟨n + 1, 0⟩ & otherwise. \end{cases}

Furthermore put, for m < f(n),

𝕀_{n,m} = \{n\} × [\frac{m}{f(n)}, \frac{m + 1}{f(n)}] .

We define a filter F_{L} on K_{f} as follows:

F_{L} = {P : L ⊆ cl_{β𝕄} ⋃_{⟨n,m⟩∈P} 𝕀_{n,m}}.

This filter has a base consisting of sets of the form

[g, h, U] = {⟨n, m⟩ : g(n) ⩽ m ⩽ h(n), n ∈ U},

where g and h are such that g/f ∈ A_{L}, h/f ∈ B_{L} and U ∈ u. From this it follows that for every subset P of K_{f}

P ∈ F_{L}  iff  σ[P] ∈ F_{L}.

We can now describe F; it is the set ⋂_{P ∈F_{L}} P^{∗} in K_{f}^{∗}. Then L = q^{∗}[𝕄_{F}].

An alternative description of F is as follows. First we consider the map π : K_{f} → ω defined by π(n, m) = n and its Čech-Stone extension π : βK_{f} → βω. The set T_{u} = π^{←}(u) contains a dense set of isolated points: Those determined by graphs of functions below f. For if g ∈ ^{ω}ω is such that g(n) ⩽ f(n) for all n then the closure of its graph meets T_{u} in exactly one point and this point is therefore isolated. This set of isolated points is linearly ordered by ⩽_{u} and the gaps in this order are filled — in K_{f}^{∗} — by sets like F. One might call F a layer in the space T_{u}.

4.2. A new layer. We shall now construct, assuming CH, an ultrafilter u, a function f and a layer F in T_{u} such that F is minimally σ-invariant. The corresponding continuum C_{F} is then a layer in 𝕀_{u}

The following Lemma provides a way to recognize minimal σ-invariant sets.

Lemma 4.1. A closed and σ-invariant set F is minimal iff for every open set that meets F there are finitely many translates that cover F.

Proof. If O is open and meets the minimal set F then the family {σ^{n}[O] : n ∈ ℤ} covers F, because every orbit is dense and meets O. As F is compact this family has a finite subcover.

Conversely, if F is not minimal and G is a proper closed σ-invariant subset of F then ω^{∗} \ G is an open set that meets F and is equal to all of its translates.

Using this Lemma we can describe what we need to construct. We must find an ultrafilter u and a gap ⟨F, G⟩ in ^{ω}ω/u below some function h, with the following property: if X ⊆ K_{h} meets every set of the form [f, g, U] with f ∈ F, g ∈ G, U ∈ u then there are f ∈ F, g ∈ G, U ∈ u and k ∈ ω such that [f, g, U] ⊆ ⋃_{|i|⩽k} σ^{i}[X]. Here, as above

[f, g, U] = {⟨n, m⟩ : f(n) ⩽ m ⩽ g(n), n ∈ U}

for f, g ∈ ^{ω}ω and U ⊆ ω.

The point is that the family of sets [f, g, U]^{∗} is a neighbourhood base for the set F, so that the previous paragraph expresses the condition of Lemma 4.1 in terms of the natural base for K_{h}^{∗}.

The desired gap may be constructed, using CH, in an induction of length ω_{1}. One constructs F = {f_{α} : α < ω_{1}}, G = {g_{α} : α < ω_{1}} and {U_{α} : α < ω_{1}} such that

(1) {U_{α} : α < ω_{1}} is decreasing mod finite and generates a P -point u.

(2) The pair ⟨F, G⟩ is a gap below the identity function Id in ^{ω}ω/u.

To take care of our condition and to ensure that we get a gap we enumerate the family of subsets of K_{h} as {X_{α} : α < ω_{1}} and ^{ω}ω as {h_{α} : α < ω_{1}}.

We start the induction by letting f_{0} be identically 0, g_{0} = Id and U_{0} = ω. We make sure that for every α one has

(1) sup{g_{α}(n) − f_{α}(n) : n ∈ U_{α}} = ω (to keep the induction alive).

(2) For all n ∈ U_{α+1} either h_{α}(n) < f_{α+1}(n) or h_{α}(n) > g_{α+1}(n) (to create a gap).

(3) If no finite number of translates of X_{α} covers [f_{α}, g_{α}, U_{α}] then X_{α} is disjoint from [f_{α+1}, g_{α+1}, U_{α+1}] (to meet our condition for minimality).

All steps are routine except when we have to take care of (3).

Assume no finite number of translates of X_{α} covers the set [f_{α}, g_{α}, U_{α}] and consider for each n ∈ U_{α} the longest interval I_{n} in {⟨n, m⟩ : m ⩽ h(n)} that is contained in [f_{α}, g_{α}, U_{α}] \ X_{α}. The claim is that sup_{n∈U_{α}} |I_{n}| = ω. For if the supremum equals k < ω then ⋃_{|i|⩽k} σ^{i}[X_{α}] contains [f_{α}, g_{α}, U_{α}]. This enables us to find functions p and q between f_{α} and g_{α} such that [p, q, U_{α}] ∩ X_{α} = ∅.

Further modifications of p, q and U_{α} will be necessary so as to satisfy (2) and to ensure that the U_{β} will in the end generate an ultrafilter.

4.3. The four extra continua. It is clear that we can construct infinitely many K_{10}-like layers in 𝕀_{u} just by varying the gap. This implies that we can take four layers L_{1}, L_{2}, L_{3} and L_{4} in 𝕀_{u} such that [0_{u}, L_{1}) and [0_{u}, L_{3}) have countable cofinality, L_{2} and L_{4} are like K_{10} and L_{1} <_{u} L_{2} <_{u} L_{3} <_{u} L_{4}.

We claim that the none of the four intervals [0_{u}, L_{2}], [L_{1}, L_{2}], [L_{2}, L_{3}] and [L_{2}, L_{4}] is homeomorphic to one of the continua K_{1} through K_{10}.

As the intervals are nontrivial and decomposable they are different from K_{1}, K_{8}, K_{9} and K_{10}. They are also different from K_{2} through K_{7}: none of these have an end set with a proper dense F_{σ}-subset.

Finally, that the intervals are mutually nonhomeomorphic can be seen by looking at the ‘other’ end set: one point (0_{u}), not a G_{δ}-set (L_{1}), a G_{δ}-set (L_{3}) and not a G_{δ}-set but not homeomorphic to L_{1} (L_{4}).

4.4. Six extra continua from ¬CH. If CH fails then Theorem ?? from Dow [1984] implies that there are u and v in ω^{∗} such that

• if L is a layer in 𝕀_{u} such that [0_{u}, L) has countable cofinality then (L, 1_{u}] has coinitiality ω_{1}, and

• if L is a layer in 𝕀_{v} such that [0_{v}, L) has countable cofinality then (L, 1_{v}]

has coinitiality larger than ω_{1}. So, if we use 𝕀_{u} to make K_{2} through K_{7} and 𝕀_{v} to make the corresponding K′_{2} through K′_{7} then the K_{i} are mutually nonhomeomorphic, the K′_{i} are mutually nonhomeomorphic, and no K_{i} is homeomorphic to any K′_{j} because K_{i} has layers of type ⟨ω, ω_{1}⟩ whereas K′_{j} has none.

5. Concluding remarks

We have, in ZFC, ten ‘honest’ different subcontinua of ℍ^{∗}. Under CH we can produce four more and under ¬CH even six more. Thus, ZFC proves that ℍ^{∗} has at least fourteen different subcontinua.

This should not be the best we can get. One would expect, given the way things usually go with the Čech-Stone compactification, to find 2^{𝔠} different subcontinua in ℍ^{∗} but at present this number seems very hard to reach.

For instance, in Dow and Hart [1993] the authors have shown that under CH all standard subcontinua of ℍ^{∗} are homeomorphic, as are all layers that determine an interval with countable cofinality.

Although trying to vary minimal σ-invariant subsets of ω^{∗} may look promising at first, one should realize that in [1990] Balcar and Błaszczyk showed that all such sets are all homeomorphic to the absolute of the Cantor cube of weight 𝔠

This leads to the (depressing) conjecture that under CH all versions of K_{10} are homeomorphic.

We finish with a question that, if answered positively, would give us five more intervals.

Question 5.1. Can the K_{9}-type continua be (homeomorphic to) layers of standard subcontinua?

References

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[1971] An non-metric indecomposable continuum. Duke Mathematical Journal, 38, 15–20. van Douwen, E. K.

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[1984] On ultra powers of Boolean algebras. Topology Proceedings, 9, 269–291. Dow, A. and K. P. Hart.

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[1986] The subcontinua of β[0, ∞) − [0, ∞). Topology Proceedings, 11, 385–413. Erratum:

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[1993] Continua in ℝ^{∗}. Topology and its Applications, 50, 183–197.

Department of Mathematics, York University, 4700 Keele Street, North York, Ontario, Canada M3J 1P3

E-mail: adow@YorkU.ca

Faculty of Technical Mathematics and Informatics, TU Delft, Postbus 5031, 2600 GA Delft, the Netherlands

E-mail: k.p.hart@twi.tudelft.nl