On representation spaces - Semantic Scholar

Topology and its Applications 164 (2014) 1–13

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Topology and its Applications www.elsevier.com/locate/topol

On representation spaces José G. Anaya a , Félix Capulín a , Enrique Castañeda-Alvarado a , Włodzimierz J. Charatonik b,∗ , Fernando Orozco-Zitli a a Universidad Autónoma del Estado de México, Facultad de Ciencias, Instituto Literario 100, Col. Centro, C.P. 50000, Toluca, Estado de México, Mexico b Department of Mathematics and Statistics, Missouri University of Science and Technology, 1870 Miner Circle, Rolla, MO 65409-0020, USA

a r t i c l e

i n f o

Article history: Received 10 January 2012 Received in revised form 13 August 2013 Accepted 24 August 2013 MSC: primary 54B80, 54C10 secondary 54A10, 54D10, 54F15

a b s t r a c t Let C be a class of topological spaces, let P be a subset of C, and let α be a class of mappings having the composition property. Given X ∈ C, we write X ∈ Clα (P) if for every open cover U of X there is a space Y ∈ P and a U -mapping f : X → Y that belongs to α. The closure operator Clα defines a topology τα in C. After proving general properties of the operator Clα , we investigate some properties of the topological space (N, τα ), where N is the space of all nondegenerate metric continua and α is one of the following classes: all mappings, confluent mappings, or monotone mappings. © 2013 Published by Elsevier B.V.

Keywords: Chainability Confluent mapping ε-Map Inverse limit Local connected Arcwise connected

1. Introduction In continuum theory many results can be expressed using ε-mappings. The subject have been started by P.S. Aleksandrov in [1], then continued by K. Kuratowski, S. Ulam, S. Eilenberg, and other in the thirties of the twentieth century (see for example [16] and [11]). Let us recall an important result by S. Marde˘sić and J. Segal (see [18]) that every continuum admits ε-mappings onto polyhedra. Note that if P is a class of ANRs the existence of ε-mappings onto members of P is equivalent to the fact that a continuum X can be expressed as an inverse limit of a sequence of elements of P with surjective bonding mappings. Therefore one can define arc-like, circle-like, tree-like, disk-like, etc., using ε-mappings * Corresponding author. E-mail addresses: [email protected] (J.G. Anaya), [email protected] (F. Capulín), [email protected] (E. Castañeda-Alvarado), [email protected] (W.J. Charatonik), [email protected] (F. Orozco-Zitli). 0166-8641/$ – see front matter © 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.topol.2013.08.012

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J.G. Anaya et al. / Topology and its Applications 164 (2014) 1–13

or, equivalently, using inverse limits. This is not true anymore if the class P does not consist of ANRs, or if we restrict the class of mappings. In fact, there is not much known about when the existence of ε-mappings is enough for the existence of an appropriate inverse limit if we restrict to some classes of mappings. The only article on the subject is [21]. Interesting results and problems can be obtained if we restrict the class of mappings to a class of confluent, monotone, or another class with composition property (see [3] or [21]). Recently, a lot of results about ε-mappings and ε-properties have been gathered in [3]. In general topology one needs to use U-mappings, for a given cover U of the domain, instead of ε-mappings. In this article we observe that one can introduce a natural closure operator and investigate the space of some topological spaces with the topology generated by that operator. This is another point of view on continuum theory or on topology in general. Some old results can be expressed in the language of the topological space (C, τα ), but also some new theorems and problems arise. 2. Definitions, notation and basic results Definition 2.1. Given two topological spaces X and Y and a cover U of X, we say that a mapping f : X → Y is a U-mapping if there is an open cover V of Y such that {f −1 (V ): V ∈ V} refines U. Definition 2.2. Let C be a class of topological spaces and let α be a class of mappings between elements of C. We say that α has the composition property if (1) for every X ∈ C the identity map idX : X → X is in α, (2) if f : X → Y and g : Y → Z are in α, then g ◦ f is in α. Definition 2.3. Let C be a class of topological spaces, let P be a subset of C, and let α be a class of mappings having the composition property. Given X ∈ C, we write X ∈ Clα (P) if for every open cover U of X there is a space Y ∈ P, and a U-mapping f : X → Y that belongs to α. Theorem 2.4. The operator Clα satisfies the following Kuratowski axioms of the closure operator (see [15, §4, p. 38]): (1) (2) (3) (4)

A ⊂ Clα (A), Clα (A) = Clα (Clα (A)), Clα (A ∪ B) = Clα (A) ∪ Clα (B), Clα (∅) = ∅.

Proof. In order to show (1), it is enough to put Y = X, to take f as the identity mapping on X, and put V = U. To show (2), first observe that (1) and (2) imply that Clα (A) ⊆ Clα (Clα (A)). To see the other inclusion, take X ∈ Clα (Clα (A)) and an open cover U of X. Then, by the definition, there is a space Y ∈ Clα (A), a mapping f : X → Y in α, and a cover V of Y such that {f −1 (V ): V ∈ V} refines U. Again, since Y ∈ Clα (A), there is a space Z ∈ A, a mapping g : Y → Z, g ∈ α, and an open cover W of Z such that {g −1 (V ): V ∈ W} refines V. To see that X ∈ Clα (A), it is enough to consider the space Z, the mapping g ◦ f , that is in α by the composition property of α, and the cover W. To show (3), first observe that (1) and (2) imply that Clα (A) ∪ Clα (B) ⊆ Clα (A ∪ B). To see the other inclusion, take X ∈ Clα (A ∪ B) \ Clα (A). We need to show that X ∈ Clα (B). Since X ∈ / Clα (A), there is an open cover U0 of X such that for every space Y ∈ A, for every f : X → Y satisfying f ∈ α, and for every cover V of Y , the family {f −1 (V ): V ∈ V} does not refine U0 . To show that X ∈ Clα (B), consider an open


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cover U of X and define U1 = {U ∩ V : U ∈ U and V ∈ U0 }. Then, there is a space Y ∈ A ∪ B, a mapping f : X → Y , f ∈ α, and an open cover V of Y such that {f −1 (V ): V ∈ V} refines U1 . Since U1 refines U0 , the set {f −1 (V ): V ∈ V} refines U0 and therefore, by our assumption, Y ∈ / A, so Y ∈ B, and the proof of (3) is complete. Finally, (4) is trivial. 2 As a consequence we see that for every set C of topological spaces and for every set α of mappings between elements of C with the composition property, we may introduce a topology τα on C. We will call τα the topology on C generated by α. As open sets we take sets P satisfying C \ P = Clα (C \ P). Note that to define the closure operator, we used topological properties of the considered sets only, so homeomorphic spaces are not distinguishable in our topological spaces. Therefore it is natural to assume that the set C does not contain two different homeomorphic spaces. Proposition 2.5. If a mapping f : X → Y between topological spaces X and Y is surjective and closed, then the following two conditions are equivalent: (1) f is a U-mapping; (2) for every y ∈ Y , there is U ∈ U such that f −1 (y) ⊆ U . Proof. The implication (1) =⇒ (2) follows from the definitions. To show (2) =⇒ (1) define, for every U ∈ U, VU = {y ∈ Y : f −1 (y) ⊆ U } and V = {VU : U ∈ U}. Then V is a cover of Y by (2) and f −1 (VU ) ⊆ U , so {f −1 (V ): V ∈ V} is a refinement of U. Finally, openness of VU for U ∈ U follows from the fact that Y \ VU = f (X \ U ) is a closed set since f is closed. 2 Proposition 2.6. If a space X is compact, P is a set of topological spaces, and α is a set of mappings with the composition property, then the following two conditions are equivalent: (1) X ∈ Clα (P); (2) for every finite open cover U of X, there is a space Y ∈ P, and a U-mapping f : X → Y that belongs to α. Proof. The implication (1) =⇒ (2) is trivial, and the implication (2) =⇒ (1) is a consequence of compactness of X. The details are left to the reader. 2 Definition 2.7. If X and Y are metric spaces and ε is a positive number, then a mapping f : X → Y is called an ε-mapping if for every y ∈ Y , we have diam(f −1 (y)) < ε. Proposition 2.8. If a space X is a metric compact space, P is a set of topological spaces, and α is a set of mappings, then the following two conditions are equivalent: (1) X ∈ Clα (P); (2) for every ε > 0, there is a space Y ∈ P, and an ε-mapping f : X → Y that belongs to α. Proof. To see the implication (1) =⇒ (2), it is enough to consider the cover U = {U : U is an open subset of X satisfying diam(U ) < ε}. Now assume (2) and let U be any open cover of X. Then, since X is compact, there is a Lebesgue number ε > 0 of the cover U, i.e. such a number that every set of diameter less then ε is contained in an element of U. Take Y ∈ P and f ∈ α as in (2). Then, by Proposition 2.5, f is a U-mapping as required. 2


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Observation 2.9. If α is a class of mappings and P is a class of spaces invariant under α (i.e. for every X ∈ P and for every f ∈ α, we have f (X) ∈ P), then P is an open property, i.e. Intα (P) = P. The proof of the following theorem follows from the definitions. Theorem 2.10. Let P be a set of topological spaces, and let α, β be sets of mappings such that α ⊂ β. Then (1) (2) (3) (4) (5)

τβ ⊂ τα , Clα (P) ⊂ Clβ (P), Intβ (P) ⊂ Intα (P), if Clβ (P) = P, then Clα (P) = P, if Intβ (P) = P, then Intα (P) = P.

Even if the definition presented above may be applied to any class of topological spaces, in the rest of the paper we will concentrate on the class N of all nondegenerate continua with different classes of mappings. Because of compactness we will use ε-mappings, not the covers. 3. All mapping Now we will investigate the space (N, τA ) of all nondegenerate continua with the topology generated by the class A of all surjective mappings. As usually when discussing a new topological space we will discuss its separation axioms, connectedness, weight, density as well as some examples of interiors and closures of some classes. Let A, F, M denote the classes of surjective mappings, confluent (surjective) mappings and monotone surjective mappings between continua, respectively. Thus, M ⊂ F ⊂ A. Theorem 3.1. The space (N, τM ) is not a T0 -space, therefore for every class α of mappings satisfying M ⊂ α, we have that (N, τα ) is not a T0 -space. Proof. First, to see that (N, τM ) is not a T0 -space, consider the dendrites Dω and D{3,4,5,...} . We will show that Dω ∈ ClM ({D{3,4,5,...} }) and D{3,4,5,...} ∈ ClM (Dω ). To show the first one, we need to find, for a given ε > 0, a monotone ε-mapping f : D{3,4,5,...} → Dω . To this aim, choose a sequence I1 , I2 , . . . of interval in D{3,4,5,...} with the following properties: (1) In ∩ Im = ∅ for n = m; (2) diam(In ) < ε for n ∈ {1, 2, 3, . . .}; (3) each ramification point belongs to In for some n ∈ {1, 2, 3, . . .}. Define f as the monotone mapping that shrinks every interval In to a point. Then the images f (D{3,4,5,...} ) is a dendrite as a monotone image of a dendrite, all ramification points of it are of order ω and the ramifications points are dense, so it is homeomorphic to Dω (see [8, Theorem 6.2, p. 229]). By condition (2), f is an ε-mapping. To see D{3,4,5,...} ∈ ClM (Dω ), we need to find, for a given ε > 0, a monotone ε-mapping f : Dω → D{3,4,5,...} . For each ramification point b, we shrink all but finitely many components of Dω \ {b} to the point {b} in such a way the diameters of the shrunk sets are less that ε. In this way the image of b has finite order. Choosing carefully the components to shrink we may ensure that the points of a given order are dense in the image. Thus the image is homeomorphic to D{3,4,..} (see [8, Theorem 6.2, p. 229]). The details are left to the reader. Finally, the conclusion for arbitrary class α containing M follows from Theorem 2.10(3). 2


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In [9, Corollary 3.5], the authors prove that the space (N, τA ) is pathwise connected, thus connected. They also show (see [9, Corollary 3.4]) that there is one element L of N such that ClA ({L}) = N, thus the density of (N, τA ) is one. The weight of the representation space was discussed during the Continuum Theory, Prague 2011: Open Problem Workshop, the participants of the discussion were: María Elena Aguilera-Miranda, Félix Capulín, Włodzimierz J. Charatonik, Luis Miguel García-Velazquez, Rodrigo Jesús Hernández-Gutiérrez, Alejandro Illanes, Verónica Martínez-de-la-Vega, Norberto Ordoñes-Ramírez, Pavel Pyrih, Javier Sánchez-Martínez, Benjamin Vejnar, and Hugo Villanueva-Méndez, but the idea of the main argument for the proof of the Theorem 3.2 below was given by Benjamin Vejnar. Here we present the result. The symbol w((X, τ )) denotes the weight of the topological space (X, τ ), the symbol χ((X, τ )) denotes the character of the space (X, τ ). Theorem 3.2. Let α be any class of mappings with the composition property. Then χ((N, τα )) ℵ0 and w((N, τα )) = c. Proof. To see that χ((N, τα )) ℵ0 , it is enough to define the set N (X, n) = {f (X): f ∈ α and f is a n1 mapping} and observe that {N (X, n): n ∈ {1, 2, . . .}} is a local base of a continuum X in the space (N, τα ). Since the cardinality of (N, τα ) is c, the base {N (X, n): X ∈ N and n ∈ {1, 2, 3, . . .}} has cardinality less than or equal to cℵ0 = c. So, w((N, τα )) c. To see that w((N, τα )) c, it is enough to observe that the set of all Waraszkiewicz spirals is a discrete subset of (N, τα ) for any α. The cardinality of the set of Waraszkiewicz spirals is c, so the argument is complete. For the definition of Waraszkiewicz spirals see [22]. 2 The following observation may be helpful in finding interiors of some classes of continua. Observation 3.3. Let α be any class of mappings with the composition property and let P be a subset of N. Then X ∈ Intα (P) if and only if there is a number ε > 0 such that for each f : X → Y , if f ∈ α and f is an ε-mapping, then Y ∈ P. Corollary 3.4. If P is an open property in (N, τA ), then “containing a subcontinuum in P” is also an open property in (N, τA ). Similarly, for closed properties we have the following observation. It was mentioned in [3]. Observation 3.5. If P is a closed property in (N, τA ), then “hereditarily P” is also a closed property in (N, τA ). Definition 3.6. Let α be a class of mappings. We say that α has the inverse limit projection property if from the fact that all bonding mappings fn in an inverse system {Xn , fn } are members of α, we may conclude that the natural projections from lim {Xn , fn } onto Xn also belong to α. ←− It is known that confluent, weakly confluent, open, and monotone maps have the inverse limit projection property (see [6]). Proposition 3.7. If α is a class of mappings with composition property and inverse limit projection property, P is a class of continua, X = lim {Xn , fn }, where Xn ∈ P and fn ∈ α, then X ∈ Clα (P). ←− Now we will give examples of interiors and closures of some classes of continua. Let us adopt the following symbols for classes of continua:


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LC HU U AL CL TR TL WT T AC K KT S CC

— — — — — — — — — — — — — —

locally connected continua, hereditarily unicoherent continua, unicoherent continua, arc-like continua, circle-like continua, trees, tree-like continua, weak triods, triods, arcwise connected continua, Kelley continua, Knaster type continua, including the arc, solenoids, including the simple closed curve, continuum chainable continua.

Most of the definitions of the classes can be found in [20] and [12], Kelley continua are called there continua with property κ; the definition of Knaster type continua can be found in [21]. All of those classes are considered as subclasses of N, i.e. all considered continua are nondegenerate. The following observation is a consequence of the definitions. Observation 3.8. (1) ClA ({arc}) = AL, (2) ClA ({simple closed curve}) = CL, (3) ClA (TR) = TL. Proposition 3.9. (1) IntA (LC) = LC, (2) ClA (LC) = N. Proof. Eq. (1) follows from Observation 2.9 and the inclusion ClA (LC) ⊂ N is obvious. To see the other one, let X ∈ N. By [18, Theorem 1 and Remark 2, p. 148], X can be represented as an inverse limit of polyhedra with surjective bonding mappings. So, by Proposition 3.7, X ∈ ClA (LC). 2 Proposition 3.10. (1) (2) (3) (4)

IntA (HU) = ∅, ClA (HU) = HU, IntA (U) = ∅, ClA (U) = U.

Proof. To see (1) and (3), we need to find, for given X ∈ N and ε > 0, a continuum Y ∈ / U and a surjective ε-mapping f : X → Y . Let U be an open subset of X with diam(U ) < ε. Choose p ∈ U and q ∈ X \ Cl(U ) such that d(p, q) < ε. Let f : X → Y be a map that shrinks Bd(U ) to a point and identifies p and q. In other words f (x) = f (y) if and only if x = y, or x, y ∈ Bd(U ), or x, y ∈ {p, q}. Then Y = f (Cl(U )) ∪ f (X \ U ) is not unicoherent, since both f (Cl(U )) and f (X \ U ) are connected and f (Cl(U )) ∩ f (X \ U ) is a two point set. Eqs. (2) and (4) were proven by K. Kuratowski and S. Ulam in [16, p. 248]. 2


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Proposition 3.11. (1) IntA (AL) = ∅, (2) ClA (AL) = AL. Proof. This is a consequence of Observation 3.8(1) and Proposition 3.10. 2 Proposition 3.12. (1) IntA (CL) = ∅, (2) ClA (CL) = CL. Proof. To see (1), notice that each proper subcontinuum of a circle-like continuum is arc-like, while the continuum Y constructed in the proof of Proposition 3.10 is not hereditarily unicoherent, thus not arc-like. Eq. (2) is a consequence of Observation 3.8(2). 2 Similarly, one can prove the following proposition. Proposition 3.13. (1) IntA (TR) = IntA (TL) = ∅, (2) ClA (TR) = ClA (TL) = TL. Proposition 3.14. (1) IntA (WT) = WT, (2) ClA (WT) = ClA (T) = N. Proof. It is easy to see that IntA (WT) ⊂ WT. To see the other inclusion, let X ∈ WT. Then there exist three subcontinua A1 , A2 , A3 of X such that 3 3 X = i=1 Ai , i=1 Ai = ∅ and, for each k ∈ {1, 2, 3}, Ak {Ai : i ∈ {1, 2, 3} \ {k}}. Given k ∈ {1, 2, 3}. Let ak ∈

Ai : i ∈ {1, 2, 3} \ {k} ,

and let rk = d ak , Ai : i ∈ {1, 2, 3} \ {k} . Let ε < min{r1 , r2 , r3 } and let Y be a continuum. For a given ε-mapping f : X → Y , we have f (ak ) ∈ / f ( {Ai : i ∈ {1, 2, 3} \ {k}}) for k ∈ {1, 2, 3}, so f (X) is the weak triod f (A1 ) ∪ f (A2 ) ∪ f (A3 ). To see (2), it is enough to find, for an arbitrary nondegenerate continuum X and number ε > 0, a surjective ε-mapping onto a triod. Let U and V be two open subsets of X satisfying Cl(U ) ∩ Cl(V ) = ∅, diam(U ) < ε, and diam(V ) < ε. Define f as a function that shrinks Bd(U ) ∪ Bd(V ) to a point. Then the image Y under f is a triod with the center point f (Bd(U ) ∪ Bd(V )). 2 Proposition 3.15. The interior IntA (T) is a proper subset of T. Proof. This is a consequence of [3, Theorem 5(j)].

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Problem 3.16. Characterize the class IntA (T). Proposition 3.17. (1) IntA (AC) = AC, (2) ClA (AC) = N. Proof. Since arcwise connectedness is a continuous invariant (see [20, 8.28, p. 133]), (1) follows from Observation 2.9. Since LC ⊂ AC, (2) follows from Proposition 3.9(2). 2 Proposition 3.18. (1) IntA (K) = LC, (2) ClA (K) = N. Proof. By Proposition 3.9(1), we have LC ⊂ IntA (K). To prove the other inclusion, we need to show that for any nonlocally connected continuum X and ε > 0, there is a surjective ε-mapping onto a non-Kelley continuum. To this aim, consider a point p ∈ X which is a point of nonconnectedness im kleinen of X. Then, there are a closed neighborhood M of p such that diam(M ) < 2 and a sequence of points pn converging to p belonging to components Cn of M different than the component C of the point p. By Theorem 5.12 of [20, p. 76], C is a nondegenerate continuum. Let q ∈ C \ {p} and δ = d(p, q). Let Y be the quotient space X/{p, q} and let f : X → Y be the quotient map. To finish the proof it is enough to show that Y is not a Kelley continuum. Denote by K the closure of the component of B(q, 4δ ) that contains the point q. Note that f (p) = f (q) ∈ f (K) and f (pn ) → f (q). If Y were a Kelley continuum, there would be continua Ln containing f (pn ) and converging to f (K). Since f is one-to-one on X \ {p} and each continuum f −1 (Ln ) contains pn , the sequence of continua {f −1 (Ln )}∞ n=1 is converging to K ∪ {p}, a contradiction. Finally, (2) is consequence of Proposition 3.9(2). 2 The proofs of the items in following proposition is left to the reader. They are consequences of previously proven propositions. Proposition 3.19. (1) (2) (3) (4) (5) (6)

IntA (KT) = ∅, ClA (KT) = AL, IntA (S) = ∅, ClA (S) = CL, IntA (CC) = CC, ClA (CC) = N.

4. Confluent mappings In this section we will investigate the space (N, τF ). By Theorem 3.1, the space (N, τF ) is not T0 . The weight w((N, τF )) is continuum by Theorem 3.2. We will see that, contrary to the previous cases, the space (N, τF ) is not connected, in fact, there are several open and closed properties, for example TL and K. Moreover, we conjecture that KT and S are such.


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Proposition 4.1. (1) IntF ({arc}) = {arc}, (2) ClF ({arc}) = KT. Proof. Since confluent image of an arc is an arc (see [20, Theorem 13.31, p. 292]), (1) follows from Observation 2.9. Eq. (2) follows from [21, Corollary 3.4, p. 120]. 2 Proposition 4.2. (1) IntF ({simple closed curve}) = {simple closed curve}, (2) ClF ({simple closed curve}) = S. Proof. Clearly, IntF ({simple closed curve}) ⊂ {simple closed curve}. To see that the other inclusion, let us recall that a confluent image of a simple closed curve is either an arc or a simple closed curve (see [20, Theorem 13.31, p. 292]). Moreover, by Proposition 3.10(4), IntA (N \ U) = N \ U, so for any simple closed curve X, there is an ε > 0 such that each ε-image of X is not unicoherent, therefore it is a simple closed curve. Eq. (2) follows from [21, Corollary 3.3, p. 120]. 2 Proposition 4.3. (1) IntF (LC) = LC, (2) ClF (LC) K. Proof. Eq. (1) follows from Theorem 2.10(5) and Proposition 3.9(1). To see that ClF (LC) ⊂ K, note that LC ⊂ K by [12, 20.4, p. 167] and ClF (K) = K by [7, Theorem 22, p. 1098]. Finally, the inequality ClF (LC) = K is a consequence of [10, Example 3.2]. 2 Problem 4.4. Characterize the class ClF (LC). Proposition 4.5. (1) (2) (3) (4)

∅ = IntF (HU) HU, ClF (HU) = HU, ∅ = IntF (U) U, ClF (U) = U.

Proof. To see that ∅ = IntF (HU) and ∅ = IntF (U), observe that being a λ-dendroid (i.e. a hereditarily unicoherent and hereditarily decomposable continuum) is a confluent invariant (see [4, XIV, p. 217]), therefore both IntF (HU) and IntF (U) contain all λ-dendroids. Thus, to show that IntF (HU) = HU and IntF (U) = U, we need to use indecomposable continua. For a given solenoid S and a number ε > 0, there is an open (and thus confluent) ε-mapping of S onto a circle, so S is neither in IntF (U) nor in IntF (HU). Finally, (2) and (4) are a consequence of Proposition 3.10(2) and (4) using Theorem 2.10(4). 2 Problem 4.6. Characterize the classes IntF (HU) and IntF (U).


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Concerning the interiors IntF (AL) and IntF (CL), it is not known if they are equal to AL and CL respectively. This problem is related to the old problem of A. Lelek if a confluent image of an arc-like continuum is arc-like and if a confluent image of a circle-like continuum is circle-like or arc-like. The proof of the following proposition follows from the fact that a confluent image of a tree is a tree, see [20, Corollary 13.43, p. 297]. Proposition 4.7. IntF (TR) = TR. Problem 4.8. Characterize the class ClF (TR). Some properties of the class ClF (TR) has been proven previously. The class ClF (TR) is properly contained in class of absolute retracts for hereditarily unicoherent continua, see [5, Theorem 3.6, p. 94], [21, Corollary 3.15, p. 126], and [10, Example 3.2]; all members of ClF (TR) are arc-Kelley continua, see [7, Theorem 2.21, p. 1103], but not all arc-Kelley dendroids are in ClF (TR), see [10, Example 3.2]. Proposition 4.9. IntF (TL) = ClF (TL) = TL, thus TL is a closed and open property, in particular (N, τF ) is not connected. Proof. The equation IntF (TL) = TL is a consequence of the fact that a confluent image of a tree-like continuum is tree-like, see [19]. The equation ClF (TL) = TL follows from Proposition 3.13(2). 2 Proposition 4.10. (1) IntF (WT) = WT, (2) ClF (WT) = N, (3) ClF (T) = N. Proof. Eq. (1) is a consequence of Proposition 3.14. To see (2) and (3), we will show that all hereditarily indecomposable continua are neither in ClF (WT) nor in ClF (T). This is because a confluent image of a hereditarily indecomposable continuum is hereditarily indecomposable, see for example, [17, (8.10), p. 72]. 2 Problem 4.11. Which of the following equations are true (1) ClF (WT) = WT, (2) IntF (T) = T, (3) ClF (T) = T? The proof of the following proposition follows from Proposition 3.17. Proposition 4.12. IntF (AC) = AC. Problem 4.13. Characterize the class ClF (AC). Proposition 4.14. (1) IntF (K) = K, (2) ClF (K) = K,


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(3) ClF (KT) = KT, (4) ClF (S) = S. Proof. Eq. (1) follows from the fact that confluent image of a Kelley continuum is a Kelley continuum, see [23, Theorem 4.3, p. 296]. Eq. (2) is shown in [7, Theorem 2.2, p. 1096]. Eqs. (3) and (4) were proven in [21, Corollaries 3.3 and 3.4, p. 120]. 2 Note that (1) and (2) of Proposition 4.14 imply that K is a closed and open property, one more argument for nonconnectedness of (N, τF ). The following problem has been asked (in different language) by P. Krupski in [14, Question, p. 83]. Problem 4.15. Is it true that IntF (KT) = KT and IntF (S) = S? If not, characterize the classes IntF (KT) and IntF (S). 5. Monotone mappings In this section we will investigate the space (N, τM ). The space is not T0 as shown in Theorem 3.1, the weight of it is c by Theorem 3.2. It is not connected, because the space (N, τF ) is not connected; we will see that (N, τM ) has even more closed and open properties, for example {arc}, {simple closed curve}, LC, HU, and U. We will need one more denotation: denote by D the class of dendrites. Proposition 5.1. (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24)

IntM ({arc}) = {arc}, ClM ({arc}) = {arc}, Int({simple closed curve}) = {simple closed curve}, ClM ({simple closed curve}) = {simple closed curve}, IntM (LC) = LC, ClM (LC) = LC, IntM (HU) = HU, ClM (HU) = HU, IntM (U) = U, ClM (U) = U, IntM (AL) = AL, ClM (AL) = AL, IntM (CL) = CL, ClM (CL) = CL, IntM (D) = D, ClM (D) = D, IntM (TR) = TR, ClM (TR) = D, IntM (TL) = TL, ClM (TL) = TL, IntM (K) = K, ClM (K) = K, IntM (KT) = KT, ClM (KT) = KT,

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(25) IntM (S) = S, (26) ClM (S) = S. Proof. Eqs. (1), (3), (5), (8), (10), (12), (14), (17), (19)–(22), (24), and (26) are consequences of their analogs for confluent mappings. Eqs. (2), (4), and (6) were proven in [3, Theorem 9]. Eqs. (7) and (9) follow from the facts that hereditary unicoherence and unicoherence are monotone invariants, see [17, (7.6), p. 59] and [20, Corollary 13.35, p. 294]. Eqs. (11) and (13) are consequences of the facts that arc-likeness and circle-likeness are monotone invariants, see [2, Theorem 3, p. 47]. To see (15) and (16), note that D = LC ∩ HU and thus the equations follow from (5) to (8) above. The inclusion ClM (TR) ⊂ D in (18), is a consequence of (16). The opposite inclusion follows from the fact that every dendrite admits monotone ε-retractions onto trees, see [20, Theorem 10.27, p. 176]. Eqs. (23) and (25) are consequences of stronger results that being a particular Knaster type continuum is monotone invariant, see [13, Theorem 4, p. 49], as well as being a particular solenoid is monotone invariant, see [13, Theorem 5, p. 51]. 2 Proposition 5.2. ClM (AC) ⊂ CC. Proof. Let X ∈ ClM (AC). We need to show that X ∈ CC. To this aim, let ε > 0, p, q ∈ X. We need to find continua C1 , C2 , . . . , Cn such that: (1) p ∈ C1 and q ∈ Cn ; (2) diam(Ci ) < ε for each i ∈ {1, 2, . . . , n}; (3) Ci ∩ Cj = ∅ if and only if |i − j| 1. Since X ∈ ClM (AC), there exist an arcwise connected continuum Y and a monotone ε-mapping f : X → Y . By Proposition 2.8 and Theorem 2.4 there is δ > 0 such that preimage of any set of diameter less than δ has diameter less than ε. Let I1 , I2 , . . . , In be subintervals of the arc joining f (p) and f (q) in Y such that: (1) f (p) ∈ I1 and f (q) ∈ In ; (2) diam(Ii ) < δ for each i ∈ {1, 2, . . . , n}; (3) Ii ∩ Ij = ∅ if and only if |i − j| 1. Then it is enough to define Ci = f −1 (Ii ) for each i ∈ {1, 2, . . . , n}.

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The inclusion in Proposition 5.2 is proper i.e. ClM (AC) = CC. The result was obtained during the Continuum Theory, Prague 2011: Open Problem Workshop, the participants of the discussion were: María Elena Aguilera-Miranda, Félix Capulín, Włodzimierz J. Charatonik, Luis Miguel García-Velazquez, Rodrigo Jesús Hernández-Gutiérrez, Alejandro Illanes, Verónica Martínez-de-la-Vega, Norberto Ordoñes-Ramírez, Pavel Pyrih, Javier Sánchez-Martínez, Benjamin Vejnar, and Hugo Villanueva-Méndez, but the idea of the main argument for the example was given by Benjamin Vejnar. The article is under preparation. References [1] P.S. Aleksandrov, Über den allgemeinen Dimensionsbegriff und seine Beziehungen zur elementen geometrischen Anschauung, Math. Ann. 98 (1928) 617–636. [2] R.H. Bing, Concerning hereditarily indecomposable continua, Pac. J. Math. 1 (1951) 43–51. [3] F. Capulín, R. Escobedo, F. Orozco-Zitli, I. Puga, On ε-properties, in: D.N. Georgiou, S.D. Iliadis, I.E. Kougias (Eds.), Selected Papers of the 2010 International Conference on Topology and Its Applications, Technological Educational Institute of Messolonghi, Patras, Greece, 2012, pp. 54–70. [4] J.J. Charatonik, Confluent mappings and unicoherence of continua, Fundam. Math. LVI (1964) 213–220.


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[5] J.J. Charatonik, W.J. Charatonik, J.R. Prajs, Hereditarily unicoherent continua and their absolute retracts, Rocky Mt. J. Math. 34 (2004) 83–110. [6] J.J. Charatonik, W.J. Charatonik, On projections and limits mappings of inverse system on compact spaces, Topol. Appl. 16 (1983) 1–9. [7] J.J. Charatonik, W.J. Charatonik, J.R. Prajs, Confluent mappings and arc Kelley Continua, Rocky Mt. J. Math. 38 (2008) 1091–1115. [8] W.J. Charatonik, A. Dilks, On self-homeomorphic space, Topol. Appl. 55 (1994) 215–238. [9] W.J. Charatonik, M. Insall, J.R. Prajs, Connectedness of the representation space for continua, Topol. Proc. 40 (2012) 331–336. [10] W.J. Charatonik, J.R. Prajs, Maps of absolute retracts for tree-like continua, Houst. J. Math. 37 (3) (2011) 967–976. [11] S. Eilenberg, Sur les transformations à petites tranches, Fundam. Math. 30 (1938) 92–95. [12] A. Illanes, S.B. Nadler Jr., Hyperspaces, Marcel Dekker, New York, 1999. [13] P. Krupski, Solenoids and inverse limits of sequences of arcs with open bonding maps, Fundam. Math. 120 (1984) 41–52. [14] P. Krupski, Open images of solenoids, in: Topology, Leningrad, 1982, in: Lect. Notes Math., vol. 1060, Springer, Berlin, 1984, pp. 76–83. [15] K. Kuratowski, Topology, vol. I, Academic Press/PWN, New York and London/Warszawa, 1966. [16] K. Kuratowski, S. Ulam, Sur un coefficient lié aux transformations continues d’ensembles, Fundam. Math. 20 (1933) 244–253. [17] T. Maćkowiak, Continuous mappings on continua, Diss. Math. 158 (1979) 1–91. [18] S. Mardešić, J. Segal, ε-mappings onto polyhedra, Trans. Am. Math. Soc. 109 (1963) 146–164. [19] B. McLean, Confluent images of tree-like curves are tree-like, Duke Math. J. 39 (1972) 465–473. [20] S.B. Nadler Jr., Continuum Theory, Marcel Dekker, New York, 1992. [21] L.G. Oversteegen, J.R. Prajs, On confluently graph-like compacta, Fundam. Math. 178 (2003) 109–127. [22] Z. Waraszkiewicz, Une famille indénombrable de continus plans dont aucun n’est l’image d’un autre, Fundam. Math. 18 (1932) 118–137. [23] R.W. Wardle, On a property of J.L. Kelley, Houst. J. Math. 3 (1977) 291–299.