1. Introduction

Recently, the study of the (n, m)−fold hyperspace suspension of continua has been ad-dressed in [1], [5], [6], [8]-[10], [14], [15], [17]-[19], [21], [22], [24].

A continuum is a nondegenerate compact connected metric space. A subcontinuum is a continuum contained in a continuum X. The set of positive integers is denoted by N.

Given a continuum X and n ∈ N, we consider the following hyperspaces of X:

All these hyperspaces are metrized by the Hausdorff metric H [11, Theorem 2.2]. The hyperspaces F_n (X) and C_n (X) are called the n−fold symmetric product of X and the n−fold hyperspace of X, respectively, we will write C(X) instead of C₁(X). It is impor-tant to note that whenever X is a continuum, all these hyperspaces are continua (see [19, 1.8.8, 1.8.9, 1.8.12]).

Let X be a continuum and let n, m ∈ N be such that m ≤ n. In 1979 Sam B. Nadler, Jr. introduced the hyperspace suspension of a continuum X as the quotient space C(X)/F₁(X) obtained from C(X) by shrinking F₁(X) to a one-point set with the quotient topology, denoted by HS(X), see [24]. Later, in 2004 Sergio Macías in-troduced the n−fold hyperspace suspension of a continuum X as the quotient space C_n (X)/F_n (X), denoted by HS_n (X), see [17]. Afterward in 2008, Juan Carlos Macías introduced the n−fold pseudo-hyperspace suspension of a continuum X as the quotient space C_n (X)/F₁(X), denoted by P HS_n (X), see [15]. Recently, in 2018 José G. Anaya, David Maya, and Francisco Vázquez-Juárez introduced the (n, m)−fold hyperspace sus-pension of X as the quotient space C_n (X)/F_m (X) obtained from C_n (X) by shrinking F_m (X) to a one-point set with the quotient topology, denoted by HS_mⁿ (X), see [1]. The fact that HS_mⁿ (X) is a continuum follows from [25, Theorem 3.10]. The study of (n, m)−fold hyperspace suspension is, therefore, a generalization of the latter research.

The main topics of this paper are summed up in the following general problem.

Problem 1. Given a continuum X and n, m ∈ N satisfying that m ≤ n, is there a topological property P that holds on HS_mⁿ (X)?

Related to Problem 1, the aim of this paper is to prove that:

(a) If X is a continuum and n, m ∈ ℕ with m ≤ n, then HS_mⁿ (X) contains an n−cell (see Theorem 3.1).

(b)If X is a continuum and n, m ∈ ℕ with m ≤ n, then HS_mⁿ (X) has property (b) (see Theorem 3.4).

(c)If X is a continuum and n, m ∈ ℕ with m ≤ n, then HS_mⁿ (X) is unicoherent (see Theorem 3.5).

(d)If X is a continuum and n, m ∈ ℕ with m ≤ n, then HS_mⁿ (X) is colocally connected (see Theorem 3.6).

(e)If X is a continuum and n, m ∈ ℕ with m ≤ n, then HS_mⁿ (X) is aposyndetic (see Corollary 3.7).

(f)If X is a continuum and n, m ∈ ℕ with m ≤ n, then HS_mⁿ (X) is finitely aposyndetic (see Theorem 3.8).

It is important to notice that those results that give a solution to Problem 1 are indeed generalizing Theorems 3.7, 4.1, and 4.2 as well as Corollary 4.3 and 4.4 proved by S. Macías in [17], respectively.

On the other hand, we present two results related to finite-dimensional Cantor manifolds, see Theorem 3.9 and Theorem 3.10.

2. Definitions and preliminary results

In this section, we present several results (with their references) that will be useful through this paper.

Given a subset A in a metric space X, int_X (A) denotes the interior of A in X. If d is the metric of a continuum X, ε > 0, A ⊂ X, and a ∈ X, then the set {x ∈ X: d(a, x) < ε} is denoted by Bd(a, ε), or B(a, ε) when there is no possibility of confusion. Let N(ε, A) =∪ {B(a, ε) : a ∈ A}. Given subsets U₁, . . . , U_r of X, with r, n ∈ ℕ, let

U₁, . . . , U_{r n} = {A ∈ C_n (X) : A ⊂ U₁∪ · · · ∪ U_r and A ∩ U_i # ∅, for each i ∈ {1, . . . , r}}.

It is known by [11, Theorem 1.2] that the family of all sets of the form U₁, . . . , U_{r n} , where r ∈ ℕ and each U_i is an open subset of X, is a basis for the topology in C_n (X), known as Vietoris topology.

Recall that a useful tool in the theory of hyperspaces is the existence of order arcs. Given a continuum X, an order arc in 2^X is a continuous function α: [0, 1] → 2^X such that α(s) ⊊ α(t), for each s, t ∈ [0, 1] with s < t. If A, B ∈ 2^X satisfy that α(0) = A and α(1) = B, then we say that α is an order arc from A to B.

Lemma 2.1.[23, (1.8)] Let A, B ∈ 2^Xbe such that A ≠ B. Then, the following two statements are equivalent:

(a)there exists an order arc in 2^Xfrom A to B,

(b)A ⊂ B and each component of B intersects A.

An arc is any space homeomorphic to [0, 1]. Given n ∈ ℕ, an n−cell is a space which is homeomorphic to [0, 1]ⁿ . A continuum is said to be decomposable provided it can be written as the union of two of its proper subcontinua.

Lemma 2.2.[16, Theorem 3.4] Let X be a continuum and n ∈ ℕ. Then, C_n (X) contains an n−cell.

Lemma 2.3.[16, Theorem 3.5] Let X be a continuum and n ∈ N. If X contains n pairwise disjoint decomposable subcontinua, then C_n (X) contains a 2n−cell.

Lemma 2.4.[7, Proposition 1(a), p. 798] Let X be a continuum and n ∈ ℕ. If V ⊂ X is an n−cell and U is an open set in X such that U ∩ V # ∅, then there is an n−cell T ⊂U∩V.

Recall that, as in [4, p. 16], let A, B be two sets with equivalence relations R and S, respectively. A function f: A → B is said to be relation-preserving provided that aRa^′ implies f(a)Sf(a^′ ).

Lemma 2.5.[4, Theorem 4.3, p. 126] Let X, Y be spaces with equivalence relations R and S, respectively, and let f: X → Y be a relation-preserving, continuous function. Then, passing to the quotient, the function f_∗: X/R → Y /S is also continuous.

A continuum X has the property (b) provided that each continuous function from X into the unit circle S¹ is homotopic to a constant function.

We say that a continuum X is unicoherent provided that for each pair A and B of subcontinua of X such that X = A ∪ B, A ∩ B is connected.

Lemma 2.6.[16, Theorem 4.7] Let X be a connected metric space. If X has the property (b), then X is unicoherent.

Lemma 2.7.[16, Theorem 4.8] Let X be a continuum and n ∈ ℕ. Then, C_n (X) has the property (b). In particular, we have that C_n (X) is unicoherent.

Lemma 2.8.[11, Theorem 19.7] If a continuum is contractible with respect to S¹, then the continuum is unicoherent.

A continuum is said to be colocally connected when each one of its points has a local base of open sets whose complement is connected.

The continuum X is aposyndetic if for each pair of points x and y of X, there exists a subcontinuum W of X such that x ∈ int_X (W ) ⊂ W ⊂ X − {y }. A continuum X is finitely aposyndetic provided that for each finite subset F of X and each point x ∈ X −F , there exists a subcontinuum W of X such that x ∈ int_X (W ) ⊂ W ⊂ X − F .

Lemma 2.9.[2, Corollary 1] If X is an unicoherent and aposyndetic continuum, then X is finitely aposyndetic.

We use the following notations: dim[X] stands for the dimension of X, dim_p [X] stands for the dimension of X at the point p ∈ X, as in [26, p. 5].

Lemma 2.10.[6, Theorem 3.1] If X is a finite-dimensional continuum and n, m ∈ N with m ≤ n, then dim[C_n (X)] is finite if and only if dim[HS_mⁿ (X)] is finite. Moreover, if either dim[C_n (X)] is finite or dim[HS_mⁿ (X)] is finite, then dim[C_n (X)] = dim[HS_mⁿ (X)].

Lemma 2.11.[13, Theorem 2.1] If X is a continuum such that dim[X] = 2, then dim[C(X)] is infinite.

Lemma 2.12.[11, Theorem 72.5] If X is a continuum such that dim[X] ≥ 3, then dim[C(X)] is infinite.

Lemma 2.13.[3, Lemma 3.1] If X is a finite-dimensional continuum and n ∈ N, then dim[F_n (X)] ≤ n · dim[X].

Lemma 2.14.[26, 7.3] Let X, Y, Z be separable metric spaces such that X = Y ∪ Z, where dim[Y ] ≤ n and dim[Z] ≤ n. If at least one of Y and Z is closed in X, then dim[X] ≤ n.

A finite-dimensional continuum X is a Cantor manifold if for any subset A of X such that dim[A] ≤ dim[X] − 2, then X − A is connected.

Lemma 2.15.[20, Theorem 4.6] The hyperspaces C_n ([0, 1]) and C_n (S¹) are 2n−dimensional Cantor manifolds, for each n ∈ ℕ.

A continuous function between continua X and Y is said to be monotone provided that point inverses are connected (equivalently if the inverse image of each subcontinuum of y is connected).

For a continuum X and n, m ∈ ℕ satisfying that m ≤ n, the symbol q_X^(n,m) denotes the natural projection q_X^(n,m): C_n (X) → HS_mⁿ (X), and F_X^m denotes the element of q_X^(n,m) (F_m (X)). Notice that

is a homeomorphism.

We shall make use of other concepts not defined here, which will be taken as in [19].

3. Main Results

The following result extends [17, Theorem 3.7].

Theorem 3.1.Let X be a continuum and n, m ∈ ℕ with m ≤ n. Then, HS_mⁿ (X) contains an n−cell.

Proof. By Lemma 2.2, C_n (X) contains an n−cell M. Moreover, since C_n (X) −F_m (X) isa dense open subset of C_n (X), we have that ((C_n (X)−Fm(X))∩M̸= ∅. By Lemma 2.4, there exists an n−cell N such that N ⊂ Cn(X)−Fm(X). Thus, by (1), HSⁿ_m (X) contains an n−cell.

The next result extends [17, Theorem 3.8].

Theorem 3.2.If n, m ∈ N with m ≤ n and X is a continuum that contains n pairwise disjoint decomposable subcontinua, then HS_mⁿ (X) contains a 2n−cell.

Proof. By Lemma 2.3, C_n (X) contains a 2n−cellM. Moreover, since C_n (X)−F_m (X) is a dense open subset of C_n (X), we have that ((C_n (X)−F_m (X))∩M̸= ∅. By Lemma 2.4, there exists a 2n−cell N such that N ⊂ C_n (X)−F_m (X). Thus, by (1), HSⁿ_m (X) contains a 2n−cell.

The following result extends [18, Theorem 4.1].

Theorem 3.3.Let X be a continuum and n, m, s ∈ ℕ with m ≤ s < n. Then, HS_m^s (X) can be embedded in HS_mⁿ (X).

Proof. Let i_s,n: C_s (X) → C_n (X) be the inclusion function, q_X^(s,m): C_s (X) → HS_m^s (X) and q_X^(n,m): C_n (X) → HS_mⁿ (X) be quotient functions. We denote q_X^(s,m) (F_m (X)) = F_X^(s,m) and q_X^(n,m) (F_m (X)) = F_X^(n,m) . Since

are partitions of C_n (X) and C_s (X), respectively; then i_s,n is a relation-preserving and continuous. Now, let h_s,n: HS_m^s (X) → HS_mⁿ (X) be given by

Notice that h_s,n is a continuous function by Lemma 2.5. Moreover, as h_s,n is defined, it is clear that h_s,n is a one-to-one function. Since the spaces are compact, h_s,n is an embedding.

The next result extends [17, Theorem 4.1].

Theorem 3.4.Let X be a continuum and n, m ∈ ℕ with m ≤ n. Then, HS_mⁿ (X) has property (b).

Proof. Let A ∈ HS_mⁿ (X). If A = F_X^m , then (q_X^(n,m) )⁻¹ ( A) = F_m (X) which is a connected subset of C_n (X). On the other hand, if A # F_X^m , using relation (1), then (q_X^(n,m) )⁻¹ (A) is a one-point set. Hence, (q_X^(n,m) )⁻¹ (A) is a connected subset of C_n (X). Therefore, q_X^(n,m) is a monotone function. By Lemma 2.7, C_n (X) has property (b). Since q_X^(n,m) (C_n (X)) = HS_mⁿ (X) and [12, Theorem 2, p.434], we conclude that HS_mⁿ (X) has the property (b).

Theorem 3.5.Let X be a continuum and n, m ∈ ℕ with m ≤ n. Then, HS_mⁿ (X) is unicoherent.

Proof. Applying Theorem 3.4 and Lemma 2.6, the result follows.

The following result extends [17, Theorem 4.2].

Theorem 3.6.Let X be a continuum and n, m ∈ ℕ with m ≤ n. Then, HS_mⁿ (X) is colocally connected.

Proof. Case n = m = 1 is already proved in [5, Theorem 4.1].

Suppose n ≥ 2 and let A ∈ HS_mⁿ (X). We are going to consider three cases:

Case 1.A = F_X^m .

For any ε > 0, let U_ε = B_H (F_m (X), ε). Notice that {q_X^(n,m) (U_ε ) : ε > 0} forms a base of open sets about F_X^m . Fix ε > 0. Let B ∈ HS_mⁿ (X) − q_X^(n,m) (U_ε ). Thus, (q_X^(n,m) )⁻¹ ( B) ∈ C_n (X) − U_ε . By Lemma 2.1, there exists an order arc α: [0, 1] → C_n (X) such that α(0) = (q_X^(n,m) )⁻¹ (B) and α(1) = X and α([0, 1]) ⊂ C_n (X) − U_ε . Notice that q_X^(n,m)◦ α: [0, 1] → HS_mⁿ (X) is an arc from B to q_X^(n,m) (X) satisfying (q_X^(n,m)◦α)([0, 1]) ⊂ HS_mⁿ (X)− (U_ε ), which implies that this space is arcwise connected.

Case 2.A = q_X^(n,m) (X).

For any ε > 0, let U_ε = B_H (X, ε). Observe that {q_X^(n,m) (U_ε ) : ε > 0} forms a base of open sets about q_X^(n,m) (X). Fix ε > 0. Let B ∈ HS_mⁿ (X) − q_X^(n,m) (U_ε ). Thus, (q_X^(n,m) )⁻¹ (B) ∈ C_n (X) − U_ε . Let D ∈ F_m ((q_X^(n,m) )⁻¹ (B)). By Lemma 2.1, there exists an order arc a : [0, 1] → C_n (X) such that α(0) = D and α(1) = (q_X^(n,m) )⁻¹ (B). Moreover, α([0, 1]) ⊂ C_n (X) − U_ε . Hence, q_X^(n,m)◦ α: [0, 1] → C_n (X) is an arc such that (q_X^(n,m)◦ α)(0) = F_X^m , (q_X^(n,m)◦ α)(1) = D and (q_X^(n,m)◦ α)([0, 1]) ⊂ HS_mⁿ (X) − q_X^(n,m) (U_ε ). Therefore, HS_mⁿ (X) − q_X^(n,m) (U_ε ) is an arcwise connected space.

Case 3.A ∈ HS_mⁿ (X) − {F_X^m, q_X^(n,m) (X)}.

For any ε > 0, let U_ε = B_H ((q_X^(n,m) )⁻¹ (A), ε). Thus, {q_X^(n,m) (U_ε ) : ε > 0} forms a base of open sets about A. Fix ε > 0 such that q_X (U_ε ) ∩ {F_X^m, q_X^(n,m) (X)} = ∅. Let B ∈ HS_mⁿ (X) − q_X^(n,m) (U_ε ). If (q_X^(n,m) )⁻¹ (B) ⊈ (q_X^(n,m) )⁻¹ (A), by Lemma 2.1 there exists an order arc α: [0, 1] → C_n (X) such that α(0) = (q_X^(n,m) )⁻¹ (B) and α(1) = X. Thus, α([0, 1]) ⊂ C_n (X)−B_H ((q_X^(n,m) )⁻¹ (A), ε). Hence, q_X^(n,m)◦α is an arc from B to q_X^(n,m) (X) such that q_X^(n,m)◦ α ⊂ HS_mⁿ (X) − U_ε , as desired.

On the other hand, suppose that (q^(n,m)_X )⁻¹ (B) ⊂ (q^(n,m)_X )⁻¹ (A). Let D ∈ F_m ((q^(n,m)_X )⁻¹ (B)). By Lemma 2.1, there exists an order arc β: [0, 1] → C_n (X) such that β(0) = D and β(1) = (q^(n,m)_X )⁻¹ (B). Thus, β([0, 1]) is contained in C_n (X) − BH((q^(n,m)_X )⁻¹ (A), ε). Hence, q(n,m) X ◦ β: [0, 1] → HSnm (X) is an arc from F^m_X to B and (q^(n,m)_X◦ β)([0, 1]) ⊂ HSⁿ_m (X) − q^(n,m)_X (U_ε ). Therefore, the last space is arcwise connected.

Since colocal connectedness implies aposyndesis, we have the next result which extends [17, Corollary 4.3].

Corollary 3.7.Let X be a continuum and n, m ∈ ℕ with m ≤ n. Then, HS_mⁿ (X) is aposyndetic.

From this, we can prove the following result which extends [17, Corollary 4.4].

Theorem 3.8.Let X be a continuum and n, m ∈ ℕ with m ≤ n. Then, HS_mⁿ (X) is finitely aposyndetic.

Proof. By Theorem 3.5, HS_mⁿ (X) is unicoherent. By Corollary 3.7, HS_mⁿ (X) is aposyn-detic. Finally, by Lemma 2.9, any aposyndetic unicoherent continuum is finitely aposyndetic.

The following result extends [17, Theorem 3.9].

Theorem 3.9.Let X be a continuum and n, m ∈ ℕ with m ≤ n. If C_n (X) is a finite- dimensional Cantor manifold such that dim[C_n (X)] ≥ n + 2, then HS_mⁿ (X) is a finite- dimensional Cantor manifold.

Proof. Let k = dim[C_n (X)]. According to Lemma 2.10, dim[HS_mⁿ (X)] = k. Suppose HS_mⁿ (X) is not a Cantor manifold. Hence, there exists a subset A of HS_mⁿ (X) such that dim[A] ≤ k−2 and HS_mⁿ (X)−A is not connected. Hence, there exist a separation A₁, A₂ of HS_mⁿ (X) − A. Furthermore, by [27, (1.4), p. 43], there exist a closed subset A^′ of A and nonempty open subsets D, E of HS_mⁿ (X) such that HS_mⁿ (X) − A^′ = D ∪ E where D ⊂ A₁ and E ⊂ A₂. Hence, C_n (X) − (q_X^(n,m) )⁻¹ (A^′ ) = (q_X^(n,m) )⁻¹ (D) ∪ (q_X^(n,m) )⁻¹ (E), where (q_X^(n,m) )⁻¹ (D) and (q_X^(n,m) )⁻¹ (E) are disjoint open subsets of C_n (X). In order to reach a contradiction, we will see that dim[(q_X^(n,m) )⁻¹ (A^′ )] ≤ k − 2 so that, C_n (X) is not a Cantor manifold. Consider two cases.

Case 1.F_X^m∈ A^′ .

Since (q_X^(n,m) )⁻¹ (A^′ ) is homeomorphic to A^′ , it follows that dim[(q_X^(n,m) )⁻¹ (A^′ )] ≤ k − 2.

Case 2.F_X^m∈ A^′ .

By Lemma 2.11 and Lemma 2.12, dim[X] = 1. Observe that (q_X^(n,m) )⁻¹ (A^′ ) = (q_X^(n,m) )⁻¹ (A^′− {F_X^m}) ∪ (q_X^(n,m) )⁻¹ ({F_X^m}) = (q_X^(n,m) )⁻¹ (A^′− {F_X^m}) ∪ F_m (X). By Lemma 2.13, dim[F_m (X)] ≤ m. Since m ≤ n ≤ k − 2 and dim[(q_X^(n,m) )⁻¹ (A^′− {F_X^m})] ≤ Lemma 2.13, dim[Fm(X)] ≤ m. Since m ≤ n ≤ k−2 and dim[(q^(n,m)_X )−1(A′ −{F^m_X})] ≤ dim[A′] ≤ k − 2, by Lemma 2.14, we conclude that dim[(q^(n,m)_X )−1(A′)] ≤ k − 2.

The following result extends [17, Corollary 3.10].

Theorem 3.10.Let n,m ∈ ℕ be such that m ≤ n. The hyperspaces HSⁿ_m ([0, 1]) and HSⁿ_m (S¹) are 2n−dimensional Cantor manifolds.

Proof. Case n = m is already proved in [17, Corollary 3.10].

Suppose that n > m. By Lemma 2.15 we have that C_n ([0, 1]) and C_n (S1) are 2n−dimensional Cantor manifolds. Since n ≥ 2 and 2n ≥ n + 2, the result follows from Theorem 3.9.

Question 3.11. For what continua X does the natural embedding in the proof of The-orem 3.3 embed HS_m^s (X) as a retract of HS_mⁿ (X)? In particular, what about the case when X is S¹?

Question 3.12. For what continua X, can HS_m^s (X) be embedded in HS_mⁿ (X) as a retract (m ≤ s < n)?

According to [6, Theorem 4.4] which states that if X is a contractible continuum and n, m ∈ ℕ with m ≤ n, then HS_mⁿ (X) is contractible, the following question arises:

Question 3.13. What continua X have the property that HS_mⁿ (X) is contractible for each n, m ∈ ℕ with m ≤ n?

Question 3.14. [6, Question 7.5] If X is decomposable and n, m ∈ ℕ with m < n, is HS_mⁿ (X) locally arcwise connected at F_X^m ?

Question 3.15. What continua X have the property that HS_mⁿ (X) is pseudo−contractible for each n, m ∈ ℕ with m ≤ n?

Acknowledgements

The authors wish to thank the referees for their valuable observations and comments, which helped to greatly improve this paper.

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Notes

Author notes

*E-mail: mjlopez@fcfm.buap.mx .