Proceedings of the Ninth Prague Topological Symposium Contributed papers from the symposium held in Prague, Czech Republic, August 19–25, 2001

A LOCALLY CONNECTED CONTINUUM WITHOUT CONVERGENT SEQUENCES

JAN VAN MILL

Abstract. We answer a question of Juhász by constructing under CH an example of a locally connected continuum without nontrivial convergent sequences.

1. Introduction

During the Ninth Prague Topological Symposium, Juhász asked whether there is a locally connected continuum without nontrivial convergent sequences. This question arose naturally in his investigation in [6] with Gerlits, Soukup, and Szentmiklóssy on characterizing continuity in terms of the preservation of compactness and connectedness. The aim of this note is to answer this question in the affirmative under the Continuum Hypothesis (abbreviated: CH).

Fedorchuk [5] constructed a consistent example of a compact space of cardinality 𝔠 containing no nontrivial convergent sequences. See also van Douwen and Fleissner [10] for a somewhat simpler construction under the Definable Forcing Axiom. These constructions yield zero-dimensional spaces. As a consequence, our construction has to be somewhat different. As in [5] and [10], we ‘kill’ all possible nontrivial convergent sequences in a transfinite process of length ω_{1}. However, our ‘killing’ is done in the Hilbert cube Q = ∏_{n=1}^{∞}[−1, 1]_{n} instead of the Cantor set.

For all undefined notions, see [4] and [11].

2000 Mathematics Subject Classification. 54A20, 54F15.

Key words and phrases. continuum, Fedorchuk space, convergent sequence, Continuum Hypothesis.

Reprinted from Topology and its Applications, in press, Jan van Mill, A locally connected continuum without convergent sequences, Copyright (2002), with permission from Elsevier Science [13].

Jan van Mill, A locally connected continuum without convergent sequences, Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp. 347–352, Topology Atlas, Toronto, 2002: arXiv:math.GN/0204145.

2. The Hilbert cube

A Hilbert cube is a space homeomorphic to Q. Let M^{Q} denote an arbitrary Hilbert cube.

A closed subset A of M^{Q} is a Z-set if for every ε > 0 there is a continuous function f: M^{Q} → M^{Q} \ A which moves the points less than ε. It is clear that a closed subset of a Z-set is a Z-set. We list some other important properties of Z-sets.

(1) Every singleton subset of M^{Q} is a Z-set.

(2) A countable union of Z-sets is a Z-set provided it is closed.

(3) A homeomorphism between Z-sets can be extended to a homeomorphism of M^{Q}.

(4) If X is compact and f: X → M^{Q} is continuous then f can be approximated arbitrarily closely by an imbedding whose range is a Z-set.

See [11, Chapter 6] for details.

Observe that by (1) and (2), every nontrivial convergent sequence with its limit is a Z-set in M^{Q}.

A near homeomorphism between compacta X and Y is a continuous surjection f: X → Y which can be approximated arbitrarily closely by homemorphisms. This means that for every ε > 0 there is a homeomorphism g: X → Y such that for every x ∈ X we have that the distance between f(x) and g(x) is less than ε.

A closed subset A ⊆ M^{Q} has trivial shape if it is contractible in any of its neighborhoods. A continuous surjection f between Hilbert cubes M^{Q} and N^{Q} is cell-like provided that f^{−1}(q) has trivial shape for every q ∈ N^{Q}. The following fundamental result is due to Chapman [3] (see also [11, Theorem 7.5.7]).

(5) Let f: M^{Q} → N^{Q} be cell-like, where M^{Q} and N^{Q} are Hilbert cubes. Then f is a near homeomorphism.

It is easy to see that if f: M^{Q} → N^{Q} is a near homeomorphism between Hilbert cubes then f is cell-like. So within the framework of Hilbert cubes the notions ‘near homeomorphism’ and ‘cell-like’ are equivalent.

A continuous surjection f between Hilbert cubes M^{Q} and N^{Q} is called a Z^{∗}-map provided that for every Z-set A ⊆ N^{Q} we have that f^{−1}[A] is a Z-set in M^{Q}.

Lemma 2.1. Let M^{Q} and N^{Q} be Hilbert cubes, and let f: M^{Q} → N^{Q} be a continuous surjection for which there is a Z-set A ⊆ M^{Q} which contains all nondegenerate fibers of f. Then f is a Z^{∗}-map.

Proof. Let B ⊆ N^{Q} be an arbitrary Z-set, and put B_{0} = B \f[A]. Write B_{0} as ⋃_{n=1}^{∞} E_{n}, where each E_{n} is compact. It follows from [11, Theorem 7.2.5] that for every n the set f^{−1}[E_{n}] is a Z-set in M^{Q}. As a consequence,

f^{−1}[B] ⊆ A ∪ ⋃_{n=1}^{∞} f^{−1}[E_{n}]

is a countable union of Z-sets and hence a Z-set by (2). □

Theorem 2.2. Let (Q_{n}, f_{n})_{n} be an inverse sequence of Hilbert cubes such that every f_{n} is cell-like as well as a Z^{∗}-map. Then

(A) \underleftarrow{lim}(Q_{n}, f_{n})_{n} is a Hilbert cube.

(B) The projection f_{n}^{∞}: \underleftarrow{lim}(Q_{n}, f_{n})_{n} → Q_{n} is a cell-like Z^{∗}-map for every n.

Proof. It will be convenient to let Q_{∞} denote \underleftarrow{lim}(Q_{n}, f_{n})_{n}.

By (5), every f_{n} is a near homeomorphism. Hence we get (A) from Brown’s Approximation Theorem for inverse limits in [2]. It follows from [11, Theorem 6.7.4] that every projection f_{n}^{∞}: Q_{∞} → Q_{n} is a near homeomorphism, hence is cell-like.

For every n let ϱ_{n} be an admissible metric for Q_{n} which is bounded by 1. The formula

ϱ(x, y) = ∑_{n=1}^{∞} 2^{−n}ϱ_{n}(x_{n}, y_{n})

defines an admissible metric for Q_{∞}. With respect to this metric we have that f_{n}^{∞} is a 2^{−(n−1)}-mapping ([11, Lemma 6.7.3]).

For (B) it suffices to prove that f_{1}^{∞} is a Z^{∗}-map. To this end, let A ⊆ Q_{1} be a Z-set, and let ε > 0. Pick n ∈ ℕ so large that 2^{−(n−1)} < ε. It follows that for every x ∈ Q_{n} we have that the diameter of the fiber (f_{n}^{∞})^{−1}(x) is less than ε. An easy compactness argument gives us an open cover U of Q_{n} such that for every U ∈ U we have that

diam(f_{n}^{∞})^{−1}[U] < ε.  (∗)

Let γ > 0 be a Lebesgue number for this cover ([11, Lemma 1.1.1]). Since f_{n}^{∞} is a near homeomorphism, there is a homeomorphism φ: Q_{∞} → Q_{n} such that for every x ∈ Q_{∞} we have

ϱ_{n}(f_{n}^{∞}(x), φ(x)) < ^{1}/_{2}γ.

Observe that A_{n} = (f_{1}^{n})^{−1}[A] is a Z-set in Q_{n}. There consequently is a continuous function ξ: Q_{n} → Q_{n} \A_{n} which moves the points less than ^{1}/_{2}γ. Now define η: Q_{∞} → Q_{∞} by

η = φ^{−1} ∘ ξ ∘ f_{n}^{∞}.

It is clear that η[Q_{∞}] misses (f_{1}^{∞})^{−1}[A]. In order to check that η is a ‘small’ move, pick an arbitrary element x ∈ Q_{∞}. By construction,

ϱ_{n}(x_{n}, ξ(x_{n})) < ^{1}/_{2}γ.

Since η(x) = φ^{−1}(ξ(x_{n})), clearly

ϱ_{n}(η(x)_{n}, ξ(x_{n})) < ^{1}/_{2}γ.

We conclude that ϱ_{n}(η(x)_{n}, x_{n}) < γ. Pick an element U ∈ U which contains both η(x)_{n} and x_{n}. By (∗) it consequently follows that ϱ(η(x), x) < ε, which is as required. □

Theorem 2.3. If (A_{n})_{n} is a relatively discrete sequence of closed subsets of Q such that \overline{⋃_{n=1}^{∞} A_{n}} is a Z-set then there are a Hilbert cube M and a continuous surjection f: M → Q such that

(A) f is a cell-like Z^{∗}-map.

(B) The closures of the sets ⋃_{n=1}^{∞} f^{−1}[A_{2n}] and ⋃_{n=0}^{∞} f^{−1}[A_{2n+1}] are disjoint.

Proof. Consider the subspace A = \overline{⋃_{n=1}^{∞} A_{n}} of Q, and the ‘remainder’ R = A \ ⋃_{n=1}^{∞} A_{n}. Observe that R is compact since the sequence (A_{n})_{n} is relatively discrete. Let T denote the product A × 𝕀; put

S = (R × 𝕀) ∪ ( ⋃_{n=1}^{∞} A_{2n} × {0}) ∪ ( ⋃_{n=0}^{∞} A_{2n+1} × {1}).

Then S is evidently a closed subspace of T. Let π: R × 𝕀 → R denote the projection. It is clear that the adjunction space (cf., [12, Page 507]) S ∪_{π} (R × 𝕀) is homeomorphic to A. By (4), any constant function S → Q can be approximated by an imbedding whose range is a Z-set. So we may assume without loss of generality that S is a Z-subset of some Hilbert cube M^{Q}. Now consider the space N = M^{Q}∪_{π}(R×𝕀) with natural decomposition map f. It is clear that f is cell-like, each non-degenerate fiber of f being an arc ([11, Corollary 7.1.2]). We will prove below that N ≈ Q. Once we know that, we also get by Lemma 2.1 that f is a Z^{∗}-map. Observe that the projection π: R × 𝕀 → R is a hereditary shape equivalence. So by a result of Kozlowski [7] (see also [1]), it follows that N is an AR. Since S is a Z-set in M^{Q} it consequently follows from [11, Proposition 7.2.12] that f[S] ≈ A is a Z-set in N. But N \ f[S] is obviously a Q-manifold, and consequently has the disjoint-cells property. But this implies that N has the disjointcells property, i.e., N ≈ Q by Toruńczyk’s topological characterization of Q in [9] (see also [11, Corollary 7.8.4]). So we conclude that f[S] ≈ A is a Z-set in the Hilbert cube N. By (3) there is a homeomorphism of pairs (Q, A) ≈ (N, f[S]). This homeomorphism may be chosen to be the ‘identity’ on A. This shows that we are done by Lemma 2.1 and the obvious fact that the sets

⋃_{n=1}^{∞} A_{2n} × {0}, ⋃_{n=0}^{∞} A_{2n+1} × {1}

have disjoint closures in M^{Q}. □

3. The construction

We will now construct our example under CH. After the preparatory work in §2, the construction is very similar to known constructions in the literature (see e.g., Kunen [8]).

Consider the ‘cube’ Q^{ω_{1}}. For every 1 ≤ α < ω_{1} let {S_{ξ}^{α} : ξ < ω_{1}} list all nontrivial convergent sequences in Q^{α} that do not contain their limits. For all α, ξ < ω_{1} pick disjoint complementary infinite subsets A_{ξ}^{α} and B_{ξ}^{α} of S_{ξ}^{α}.

We shall construct for 1 ≤ α ≤ ω_{1} a closed subspace M_{α} ⊆ Q^{α}. The space we are after will be M_{ω_{1}}.

Let τ: ω_{1} → ω_{1}×ω_{1} be a surjection such that τ(β) = ⟨α, ξ⟩ implies α ≤ β. For α ≤ β ≤ ω_{1} let π_{α}^{β} be the natural projection from Q^{β} onto Q^{α}. The

following conditions will be satisfied:

(A) M_{α} ≈ Q for every 1 ≤ α < ω_{1}, and if α ≤ β then π_{α}^{β}[M_{β}] = M_{α}.

We put ρ_{α}^{β} = π_{α}^{β} ↾ M_{β}: M_{β} → M_{α}. (B) If α ≤ β then ρ_{α}^{β}: M_{β} → M_{α} is a cell-like Z^{∗}-map. (C) If β < ω_{1}, τ(β) = ⟨α, ξ⟩, and S_{ξ}^{α} ⊆ M_{α} then (ρ_{α}^{β+1})^{−1}[A_{ξ}^{α}] and

(ρ_{α}^{β+1})^{−1}[B_{ξ}^{α}] have disjoint closures in M^{β+1}. Observe that the construction is determined at all limit ordinals γ. By compactness and (A) we must have

M_{γ} = {x ∈ Q^{γ} : (∀α < γ)(π_{α}^{γ}(x) ∈ M_{α})}.

Also, if (γ_{n})_{n} is any strictly increasing sequence of ordinals with γ_{n} ↗ γ then M_{γ} is canonically homeomorphic to

\underleftarrow{lim} (M_{γ_{n}}, ρ_{γ_{n}}^{γ_{n+1}})_{n}.

By Theorem 2.2 this implies that M_{γ} ≈ Q and also that ρ_{γ_{n}}^{γ} is a cell-like Z^{∗}-map for every n. Since γ_{1} can be any ordinal smaller than γ, the same argument yields that ρ_{α}^{γ} is a cell-like Z^{∗}-map for every α < γ. So in our construction we need only worry about successor steps.

Put M_{1} = Q^{{0}}, and let 1 ≤ β < ω_{1} be arbitrary. We shall construct M_{β+1} assuming that M_{β} has been constructed. To this end, let τ(β) = ⟨α, ξ⟩. We make the obvious identification of Q^{β+1} with Q^{β} × Q. If S_{ξ}^{α} ⊈ M_{α} then there is nothing to do. We then fix any element q ∈ Q, and put

M_{β+1} = M_{β} × {q}.

So assume that S_{ξ}^{α} ⊆ M_{α}. By Theorem 2.3 there exists a cell-like Z^{∗}-map f: Q → M_{β} such that

f^{−1}[(ρ_{α}^{β})^{−1}[A_{ξ}^{α}]], f^{−1}[(ρ_{α}^{β})^{−1}[B_{ξ}^{α}]]

have disjoint closures in Q. Put

M_{β+1} = {⟨f(x), x⟩ ∈ Q^{β} × Q : x ∈ Q}.

So M_{β+1} is nothing but the graph of f. It is clear that M_{β+1} is as required.

Now put M = M_{ω_{1}}. Observe that M is a locally connected continuum, being the inverse limit of an inverse system of locally continua with monotone surjective bonding maps (see e.g., [4, 6.3.16 and 6.1.28]). Assume that T is a nontrivial convergent sequence with its limit x in M. Since T ∪ {x} is countable, there exists α < ω_{1} such that ρ_{β}^{ω_{1}} ↾ (T ∪ {x}) is one-to-one and hence a homeomorphism for every β ≥ α. Pick ξ < ω_{1} such that S_{ξ}^{α} = ρ_{α}^{ω_{1}}[T], and β ≥ α such that τ(β) = ⟨α, ξ⟩. Then ρ_{β+1}^{ω_{1}}[T ∪ {x}] is a nontrivial convergent sequence with its limit in M_{β+1} which is mapped by ρ_{α}^{β+1} onto S_{ξ}^{α} with its limit. But this is clearly in conflict with (C).

References

[1] Fredric D. Ancel, The role of countable dimensionality in the theory of cell-like relations, Trans. Amer. Math. Soc. 287 (1985), no. 1, 1–40. MR 86b:54012

[2] Morton Brown, Some applications of an approximation theorem for inverse limits, Proc. Amer. Math. Soc. 11 (1960), 478–483. MR 22 #5959

[3] T. A. Chapman, Lectures on Hilbert cube manifolds, Regional Conference Series in Mathematics, Expository lectures from the CBMS Regional Conference held at Guilford College, October 11-15, 1975.

[4] Ryszard Engelking, General topology, second ed., Heldermann Verlag, Berlin, 1989. MR 91c:54001

[5] V. V. Fedorčuk, A compact space having the cardinality of the continuum with no convergent sequences, Math. Proc. Cambridge Philos. Soc. 81 (1977), no. 2, 177–181. MR 54 #13827

[6] J. Gerlits, I. Juhász, L. Soukup, and Z. Szentmiklóssy, Characterizing continuity by preserving compactness and connectedness, Proceedings of the Ninth Prague Topological Symposium. Contributed papers from the symposium held in Prague, Czech Republic, August 19–25, 2001, Topology Atlas, 2001, pp. 93–118.

[7] G. Kozlowski, Images of anr’s, Unpublished manuscript.

[8] Kenneth Kunen, A compact L-space under CH, Topology Appl. 12 (1981), no. 3, 283–287. MR 82h:54065

[9] H. Toruńczyk, On CE-images of the Hilbert cube and characterization of Q-manifolds, Fund. Math. 106 (1980), no. 1, 31–40. MR 83g:57006

[10] Eric K. van Douwen and William G. Fleissner, Definable forcing axiom: an alternative to Martin’s axiom, Topology Appl. 35 (1990), no. 2-3, 277–289. MR 91k:03133

[11] Jan van Mill, Infinite-dimensional topology, North-Holland Publishing Co., Amsterdam, 1989. MR 90a:57025

[12] , The infinite-dimensional topology of function spaces, North-Holland Publishing Co., Amsterdam, 2001. MR 1 851 014

[13] , A locally connected continuum without convergent sequences, To appear in Topology Appl., 2002.

Faculty of Sciences, Division of Mathematics and Computer Science, Vrije Universiteit, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands

E-mail address: vanmill@cs.vu.nl