We construct, assuming Jensen's principle diamond, a one-dimensional locally connected hereditarily separable continuum without convergent sequences. The construction is an inverse limit in omega_1 steps, and is patterned after the original Fedorchuk construction of a compact S-space. To make it one-dimensional, each space in the inverse limit is a copy of the Menger sponge.
Deep Dive into One Dimensional Locally Connected S-spaces.
We construct, assuming Jensen’s principle diamond, a one-dimensional locally connected hereditarily separable continuum without convergent sequences. The construction is an inverse limit in omega_1 steps, and is patterned after the original Fedorchuk construction of a compact S-space. To make it one-dimensional, each space in the inverse limit is a copy of the Menger sponge.
All topologies discussed in this paper are assumed to be Hausdorff. A continuum is any compact connected space. A nontrivial convergent sequence is a convergent ω-sequence of distinct points. As usual, dim(X) is the covering dimension of X; for details, see Engelking [5]. "HS" abbreviates "hereditarily separable". We shall prove:
Theorem 1.1 Assuming ♦, there is a locally connected HS continuum Z such that dim(Z) = 1 and Z has no nontrivial convergent sequences.
Note that points in Z must have uncountable character, so that Z is not hereditarily Lindelöf; thus, Z is an S-space.
Spaces with some of these features are well-known from the literature. A compact F-space has no nontrivial convergent sequences. Such a space can be a continuum; for example, the Čech remainder β[0, 1)[0, 1) is connected, although not locally connected; more generally, no infinite compact F-space can be either locally connected or HS. In [13], van Mill constructs, under the Continuum Hypothesis, a locally connected continuum with no nontrivial convergent sequences. Van Mill’s example, constructed as an inverse limit of Hilbert cubes, is infinite dimensional. Here, we shall replace the Hilbert cubes by one-dimensional Peano continua (i.e., connected, locally connected, compact metric spaces) to obtain a one-dimensional limit space. Our Z = Z ω 1 will be the limit of an inverse system Z α : α < ω 1 . Each Z α will be a copy of the Menger sponge [11] (or Menger curve) MS; this one-dimensional Peano continuum has homogeneity properties similar to those of the Hilbert cube. The basic properties of MS are summarized in Section 2, and Theorem 1.1 is proved in Section 3.
In [13], as well as in earlier work by Fedorchuk [7] and van Douwen and Fleissner [3], one kills all possible nontrivial convergent sequences in ω 1 steps. Here, we focus primarily on obtaining an S-space, modifying the construction of the original Fedorchuk S-space [6]; we follow the exposition in [4], where the lack of convergent sequences occurs only as an afterthought.
We do not know whether one can obtain Z so that it satisfies Theorem 1.1 with the stronger property ind(Z) = 1; that is, the open U ⊆ Z with ∂U zerodimensional form a base. In fact, we can easily modify our construction to ensure that 1 = dim(Z) < ind(Z) = ∞; this will hold because (as in [4]) we can give Z the additional property that all perfect subsets are G δ sets; see Section 5 for details.
We can show that a Z satisfying Theorem 1.1 cannot have the property that the open U ⊆ Z with ∂U scattered form a base; see Theorem 4.12 in Section 4. This strengthening of ind(Z) = 1 is satisfied by some well-known Peano continua. It is also satisfied by the space produced in [8] under ♦ by an inductive construction related to the one we describe here, but the space of [8] was not locally connected, and it had nontrivial convergent sequences (in fact, it was hereditarily Lindelöf).
The Menger sponge MS [11] is obtained by drilling holes through the cube [0, 1] 3 , analogously to the way that one obtains the middle-third Cantor set by removing intervals from [0, 1]. The paper of Mayer, Oversteegen, and Tymchatyn [12] has a precise definition of MS and discusses its basic properties.
In proving theorems about MS, one often refers not to its definition, but to the following theorem of R. D. Anderson [1,2] (or, see [12]), which characterizes MS. This theorem will be used to verify inductively that Z α ∼ = MS. The fact that MS satisfies the stated conditions is easily seen from its definition, but it is not trivial to prove that they characterize MS. Theorem 2.1 MS is, up to homeomorphism, the only one-dimensional Peano continuum with no locally separating points and no non-empty planar open sets.
Here, C ⊆ X is locally separating iff, for some connected open U ⊆ X, the set U \ C is not connected. A point x is locally separating iff {x} is. This notion is applied in the Homeomorphism Extension Theorem of Mayer, Oversteegen, and Tymchatyn [12]: Theorem 2.2 Let K and L be closed, non-locally-separating subsets of MS and let h : K ։ L be a homeomorphism. Then h extends to a homeomorphism of MS onto itself.
The non-locally-separating sets have the following closure property of Kline [9] (or, see Theorem 2.2 of [12]): Theorem 2.3 Let X be compact and locally connected, and let K = {K i : i ∈ ω}, where K and the K i are closed subsets of X. If K is locally separating then some K i is locally separating.
For example, these results imply that in MS, all convergent sequences are equivalent. More precisely, points in MS are not locally separating, so if x i : i ∈ ω converges to x ω , then {x i : i ≤ ω} is not locally separating. Thus, if s i and t i are nontrivial convergent sequences in MS, with limit points s ω and t ω , respectively, then there is a homeomorphism of MS onto itself that maps s i to t i for each i ≤ ω.
The following consequence of Theorem 2.1 was noted by Prajs [14] (see p. 657).
Lemma 2.4 Let J ⊆
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