Representation space with confluent mappings

Transcription

1 Representation space with confluent mappings José G. Anaya a, Félix Capulín a, Enrique Castañeda-Alvarado a, W lodzimierz J. Charatonik b,, Fernando Orozco-Zitli a a Universidad Autónoma del Estado de México, Facultad de Ciencias, Instituto Literario No. 100, Col. Centro, C. P , Toluca, Estado de México, México. b Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla MO , USA. Abstract Given a subclass P of the set N of all non-degenerate continua we say X Cl F (P) if for every ε > 0 there are a continuum Y P and a confluent ε-map f : X Y. This closure operator Cl F gives a topology τ F on the space N, see [1]. In this article we continue investigation of the topological space (N, τ F ), we establish interiors and closures of some natural classes of continua, we recall related results and pose several open problems. This gives us a new point of view on topological properties of some classes of continua and on confluent mappings. Keywords: Confluent mapping, continuum, ε-map, fan, inverse limit, representation space MSC: Primary 54B80, 54C10; Secondary 54A10, 54D10, 54F15 1. Introduction 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. 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 Corresponding author addresses: jgao@uamex.mx (José G. Anaya), fcapulin@gmail.com (Félix Capulín), eca@uaemex.mx (Enrique Castañeda-Alvarado), wjcharat@mst.edu (W lodzimierz J. Charatonik ), forozcozitli@gmail.com (Fernando Orozco-Zitli) Preprint submitted to Elsevier January 11, 2017

2 (1) for every X C the identity map id X : X X is in α, (2) if f : X Y and g : Y Z are in α, then g f is in α. 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 Cl α (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 Cl α defines a topology τ α in C. In [1] are proved general properties of the operator Cl α and many properties of the topological space (N, τ α ), where N is the space of all non-degenerate metric continua and α is one of the following classes: all mappings, confluent and monotone mappings. Readers specially interested in this topic are referred to [1], [5] and [13]. If X is a metric continuum, d denote a metric in X, d(a, b) denote the distance between the points a and b and if A, B X, dist (A, B) denote the distance between the sets A and B, defined as the infimuum of all distances d(p, q), where p A and q B. Now in this paper we will give examples of interiors and closures of some classes of continua when α is the family of confluent mappings. 2

3 2. Definitions, notation and basic results Let us adopt the following symbols for classes of continua: 3. Graphs AK arc Kelley continua, Dim1 continua of dimension 1, CF cones over 0-dimensional sets, D dendroids, D 0 dendrites, F fans (excluding the arc), G graphs, HU hereditarily unicoherent continua, K Kelley continua, KT Knaster type continua, including the arc, LC locally connected continua, λd λ-dendroids, NO n-ods, for n 3, SD smooth dendroids, SF smooth fans, S solenoids, TR trees, TL tree-like continua. Let us start with recalling results shown in [1]. Theorem Int F ({arc}) = {arc}, 2. Cl F ({arc}) = KT, 3. Int F ({simple closed curve}) = {simple closed curve}, 4. Cl F ({simple closed curve}) = S. The following theorem has been shown in [21, Corollary 3.15, p. 126]. Theorem 3.2. Cl F (G) = Cl F (LC) Dim1, in particular each locally connected continuum of dimension 1 is in Cl F (G). 3

4 Let Y be a topological space and A Y. Let β be a cardinal number. We say that A is of order less than or equal to β in Y, written ord (A) Y β, provided that for each open subset U of Y containing A, there exists an open subset V of Y such that A V U and the cardinality of the boundary of V is less than or equal to β. We say that A is of order β in Y, written ord (A) Y = β, provided that ord (A) Y β and ord (A) Y α for any cardinal number α < β. Let G be a graph and x G, x is called a branch point of G provided that ord (x) G 3. To show a result about interiors of some classes, we need the following two facts from the literature. Theorem 3.3 below is a consequence of [22, Corollary 1.7] and Theorem 3.5 was shown in Theorem of [20, p. 298]. Theorem 3.3. Confluent image of a graph is a graph. Corollary 3.4. Int F (G) = G. Theorem 3.5. (Branch Point Covering) Let X be a hereditarily arc-wise connected continuum, and let f be a confluent map of X onto a continuum Y. Then Y is hereditarily arc-wise connected, and the center of any simple triod T of Y is the image under f of the center of some simple triod T in X; also T may be chosen so that f(t ) = T. Theorem 3.6. If G is a graph, then there is an ε > 0 such that every confluent ε-image of G is homeomorphic to G. Proof. Choose an ε > 0 in such a way that the distance from any ramification points v of G to any points of a maximal free arc that does not contain v is greater than 3ε. In particular the distance between any different ramification points is greater than 3ε. Let f : G Y be a confluent ε-mapping. By Theorem 3.3 the image is a graph and by Theorem 3.5 we have that R(Y ) f(r(g)). We will show that in fact f(r(g)) = R(Y ). To obtain this result it is enough to show that the image of any ramification point is a ramification point. Let p R(G). By the choice of ε, there are three points a, b, c such that the arcs ap, bp and cp have only p as a common point, a, b, c / R(G) and the distance from a to any point of bp cp is greater than ε, similarly the distance from b to ap cp and the distance from c to bp ap are greater than ε. Moreover we may suppose that that the triod ap bp cp B(p, 2ε). Because of f is an ε-mapping, f(a) / f(bp cp), f(b) / f(ap cp) and f(c) / f(bp ap), so f(ap cp bp) is a weak triod. Since each weak triod is a triod when the weak triod is unicoherent (by Theorem of [20, p. 4

5 209], the continuum f(ap cp bp) contains a ramification point. Denote by q the ramification point of f(ap cp bp). By Theorem 3.5 there is a ramification point p such that f(p ) = q. We will show that p = p. If not, there is a point p ap cp bp such that f(p ) = q. Finally we have that d(p, p ) d(p, p ) + d(p, p ) < ε + 2ε = 3ε. This is a contradiction to the choice of ε, since p and p are different ramification points of G. To finish the proof we need to observe the equality f(r(g)) = R(Y ) implies that for any maximal arc A with end points a and b such that a, b R(G) the image is a free arc f(a) with end points f(a) and f(b). Similarly if A is a simple closed curve containing only one ramification point of G, then f(a) is a simple closed curve with only one ramification point of Y. Therefore G and Y are homeomorphic. Corollary 3.7. For every graph G we have Int F ({G}) = {G}. Theorem 3.8. If G is a graph different from the arc and from the simple closed curve and f : G G is a confluent map, then f is monotone. Proof. Consider R(G) = {x G : x is the branch point of some simple triod in G}. By Theorem 3.5, f R(G) : R(G) R(G) is a bijection. Let v R(G) and n 3. Consider a simple n-od T such that v T and T (R(G) \ {v}) =. Notice that f 1 (T ) R(G) is a one point set. In order to prove that f 1 (T ) is connected, let K be a component of f 1 (T ) such that K R(G) =. So, K is an arc. By Lemma of [20, p. 295], f K : K T is confluent mapping. Hence, by Theorem of [20, p. 295], T is an arc, a contradiction. We will prove that f 1 (v) is connected. Suppose otherwise, then there exists a small simple triod T such that v T and f 1 (T ) is not connected, a contradiction. Let x G \ R(G). We will prove that f 1 (x) is connected. First of all, notice that if A is an edge, f(int (A)) B where B is an edge of G. Therefore by Theorem of [20, p. 298] if A 1 and A 2 are different edges of G, f(int (A 1 )) f(int (A 2 )) =. Then we have a bijection between the set of edges of G. Since x G \ R(G), x we where either e R(G) or e is and end point of G and w R(G). Suppose that f 1 (x) is not connected, then there are at least two components B 1 and B 2 of f 1 (x). Since x G \ R(G), B 1 R(G) = B 2 R(G) =. Claim: B 1 and B 2 are just in one edge, if not B 1 ww 1 and B 2 ww 2, where w 1, w 2 R(G) and w 1 w 2, so f(ww 1 ) = f(ww 2 ) a contradiction to the fact that f is a bijection between 5

6 the set of edges of G. Let a, b f 1 (x) be in different components of f 1 (x). Let vv = f(w)f(e) be an edge of G such that a, b vv. Since a, b are in different components of f 1 (x), there is a point t in the arc from a to b with f(a) = f(b) f(t). We will use the orders of the intervals from a to b and from w to e. If f(t) > f(a), let m = max{f(t) : t ab}. Let t 0 ab such that f(t 0 ) = m. Since t 0 / R(G), m / R(G). Consider Q = mm me. Let C be a component of f 1 (Q) containing t 0. Notice that f(c) = {t 0 }, so f is not confluent, a contradiction. The case f(t) < f(a) can be shown similarly. Theorem 3.9. Every monotone map from a graph onto itself is a near homeomorphism. Proof. Let f : G G be a monotone map. We will show that f is a near homeomorphism. We will divide the proof in several steps. Step 1. The restriction f R(G) is a permutation of the finite set R(G). This is a consequence of Theorem of [20, p. 298] and the fact that the domain and the range of the function are the same. Step 2. If A is a free arc with end points v and w and v, w R(G), then f(a) is a free arc. Suppose not, i.e. there is a point x A such that f(x) is a ramification point different from f(v) and f(w). Then, by Step 1, there is a point y R(G) such that f(y) = f(x). By monotonicity of f the set f 1 (f(x)) is connected, it contains a point x A and a point y / A, so it has to contain either v or w, a contradiction. Step 3. If A is a free arc with end points v and w and v, w R(G) then f A is a near homeomorphism. This is a consequence of the fact that every monotone map of an interval is a near homeomorphism. Step 4. If A is simple closed curve in G such that A R(X) = {v}, then f(a) is a simple closed curve such that f(a) R(G) = {f(v)} and f A is a near homeomorphism. Similarly this is a consequence of the fact that every monotone map of simple closed curve is a near homeomorphism. Step 5. The function f is a near homeomorphism. This is a consequence of the previous steps: it is one-to-one on vertices and it is a near homeomorphism on each edge. Corollary If G is a graph different from the arc and from the simple closed curve and f : G G is a confluent map, then f is a near homeomorphism. 6

7 Theorem If G is a graph different from the arc and from the simple closed curve, then Cl F ({G}) = {G} i.e. {G} is a closed and open property. Proof. Take a continuum X Cl F ({G}) then by [21, Corollary 3.14, p. 126] X can be expressed as an inverse limit of G with confluent bonding mappings. By Corollary 3.10 the bonding mappings are near homeomorphisms. It is known that in this case we can replace the bonding mappings by homeomorphisms, so X is homeomorphic to G as required. 4. fans By a fan we mean a dendroid with exactly one ramification point. This may differ from the definition used by some other authors, as they may include the arc as a fan; for convenience we exclude this case. If X is a fan, the symbol S(X) denotes the set of end points together with the vertex of the fan. Notice that: 1. NO = TR F = G F, 2. LC F = NO {F ω }, 3. NO CF SF F. By a triod we mean a continuum T that contains a subcontinuum Z of T such that T \ Z has at least three components. Equivalently, T is a triod if there are three subcontinua A, B, and C of T such that A B = A C = B C = A B C is a proper subcontinuum of each of A, B, and C. A weak triod is a continuum T such that there are three proper subcontinua A, B, and C of T satisfying T = A B C, A B C, and no two of A, B, and C cover T. It is known that each weak triod contains a triod (see [23, Theorem 1.8]). Proposition 4.1. Let f : X Y be a confluent map between continua. If Y contains a triod, so does X. Proof. Let T Y be a triod, let T = A B C, where A, B and C are subcontinua of T such that A B = A C = B C = A B C is a proper subcontinuum of each of A, B, and C. Choose a point p f 1 (A B C) and define A, B and C as components of f 1 (A), f 1 (B) and f 1 (C) that contain the point p, respectively. We will show that T = A B C is 7

8 a weak triod in X, so it contains a triod in X. We need to show that no two of A, B, and C cover T. Really, if A B = T, then T = f(t ) = f(a B ) = f(a ) f(b ) = A B, a contradiction. The last equation is a consequence of the confluence of f. Theorem Int F (F) = F, 2. Cl F (F) = F. Proof. The equation (1) follows from the fact that a confluent image of a fan is a fan or an arc (see [3, Theorem 12, p. 32]). If ε is small enough, the confluent ε-image is a fan. To show (2), let X Cl F (F). We will list some properties of X leading to X is a fan. Property 1. X is hereditarily unicoherent. Really it follows from the fact that hereditary unicoherence is a closed property, see [1, Proposition 4.10 (2)]. Property 2. X does not contain two disjoint triods. Suppose on the contrary that T 1 and T 2 are disjoint triods in X. Then, for sufficiently small ε > 0, ε-images of T 1 and T 2 would be disjoint triods in a fan, a contradiction. Property 3. For every confluent mapping f : X Y from X onto a fan Y with a vertex v, the preimage f 1 (v) is connected. Suppose that f 1 (v) is not connected. Then there exists a small simple triod T such that v T Y, and f 1 (T ) is not connected. Then, proceeding as the proof of Proposition 4.1 one can show that each component of f 1 (T ) contains a triod, contrary to Property 2. Property 4. X contains a triod. This is a consequence of Proposition 4.1. Property 5. X is a continuum chainable continuum. 8

9 Let ε > 0 be given and let f : X Y be a confluent ε-map from X onto a fan Y with a vertex v. Then f 1 (v) is connected, choose a point p f 1 (v). It is enough to construct, for an arbitrary point a X, an ε- chain C 1, C 2,..., C n of continua such that a C 1 and p C n. To this, choose δ > 0 such that diam (f 1 (P )) < ε for each set P with diam (P ) < δ. Divide the arc f(a)v in Y into subarcs A 1, A 2,..., A n 1 satisfying diam (A i ) < δ for i {1, 2,..., n 1} and f(a) A 1. Let C 1 be the component of f 1 (A 1 ) that contains a, let C 2 be a component of f 1 (A 2 ) that intersects A 1, let C 3 be a component of f 1 (A 3 ) that intersects A 2 and so on. Note that C n 1 intersects f 1 (v). Finally we put C n = f 1 (v) and recall that C n is connected by Property 3. Then C 1, C 2,..., C n is the required chain. This finishes the proof of Property 5. Property 6. X is a fan. It was shown in [15, Lemma 1.3] that every hereditarily unicoherent and continuum chainable continuum is a dendroid, thus X is a dendroid by Properties 1 and 5. It contains exactly one ramification point by Properties 2 and 4, so it is a fan. Theorem Int F (SF) = SF, 2. Cl F (SF) = SF. Proof. The equation (1) follows from the fact that a confluent image of a smooth fan is a smooth fan (see [3, Theorem 13, p. 33]). To see (2) choose X Cl F (SF). By Theorem 4.2, X is a fan, denote by v its vertex, and suppose on the contrary that X is not a smooth fan. Then there is a sequence {a n } n=1 converging to a point a X such that lim va n exists, but va lim va n. Choose a point b lim va n \ va and choose ε > 0 satisfying ε < d(b, x) for every x in va. Let f : X Y be a confluent ε-map of X onto a smooth fan Y. Then f(v) is the vertex of Y. By [4, Theorem 4.1, p. 14], we have f(va) = f(v)f(a) and f(va n ) = f(v)f(a n ), so lim f(va n ) = lim f(v)f(a n ) = f(va) by the smoothness of Y. Since b lim va n, there is a point x 0 in va such that f(b) = f(x 0 ). Note that d(b, x 0 ) > ε by the definition of ε, thus f is not an ε-map, a contradiction. Theorem

10 1. Int F (NO) = NO, 2. Cl F (NO) = Cl F (K F) = K F, 3. Int F (K F) = K F. Proof. The equation (1) is a consequence of the fact that being a particular graph is an open property, see Theorem 3.11 (1). The inclusion Cl F (NO) Cl F (K F) is obvious, and the equation Cl F (K F) = K F and Int F (K F) = K F are consequences of the facts that both K and F are closed and open properties, see [1, Proposition 4.14] and Theorem 4.2. Finally the inclusion K F Cl F (NO) was shown in [4, Theorem 12.6, p. 59], as the equivalence of conditions 4.(a) and 5. Recall that CF can be characterized as a class of smooth fans with closed set of end-points, see [4, Theorem 12.7, p. 60]. Moreover CF K F. Theorem Int F (CF) = CF, 2. Cl F (CF) = K F. Proof. To prove (1) take X CF and choose ε < d(v, e) for any e E(X). Let f : X Y be a confluent ε-map. Then Y is a Kelley fan by Theorem 4.4 (2), so S(X) is closed. By the definition of ε, no point of E(X) is mapped onto f(v), so E(Y ) = f(e(x)) is a closed subset of Y, thus Y CF. The inclusion Cl F (CF) K F follows from Theorem 4.4 (2). To see the inclusion K F Cl F (CF), let X K F and let ε > 0. We need to construct a confluent ε-mapping onto a fan Y CF. Note that S(X) is closed, it contains no nondegerate subcontinuum and thus it is 0-dimensional. Denote by v the vertex of X and choose a closed neighbourhood U of v such that diam(u) < ε and Bd (U) E(X) =. The required function is just shrinking U to a point. Theorem Int F ({harmonic fan}) =, 2. Cl F ({harmonic fan}) = (K F) \ LC. Proof. To show (1) observe that, for every ε > 0, there is a confluent (even open) ε-retraction of the harmonic fan onto an n-odd for some n. 10

11 To see the inclusion Cl F ({harmonic fan}) (K F) \ LC choose X Cl F ({harmonic fan}). By Theorem 4.3, X is a smooth fan, denote by v its vertex. We will show that S(X) is closed. Suppose on the contrary, then there is a sequence of points {e n } n=1 in S(X) converging to a point a X such that a / S(X). Choose an end-point e E(X) such that va ve and choose ε > 0 satisfying ε < min{d(a, e), d(v, a)}. Let f : X Y be a confluent ε-map of X onto the harmonic fan Y. Then f(v) is the vertex of Y. By [4, Theorem 4.1, p. 14], we have f(va) = f(v)f(a) and f(ve n ) = f(v)f(e n ), so lim f(ve n ) = lim f(v)f(e n ) = f(va) by the smoothness of Y. Since f(e n ) S(Y ), f(a) S(Y ). Since f(va) = f(v)f(a), f(a) = f(e) by [4, Theorem 4.1, p. 14], a contradiction. This finishes the proof that S(X) is closed. Since every smooth fan with S(X) closed is Kelley, see [4, Theorem 12.3, p. 58], we have X K F. It cannot be locally connected, because local connectedness is a continuous invariant. This finishes the proof that Cl F ({harmonic fan}) (K F) \ LC. To see that (K F) \ LC Cl F ({harmonic fan}), let X (K F) \ LC and let ε > 0. We need to construct a confluent ε-mapping from X onto the harmonic fan Y. In the first step we will construct a monotone ε-map from X onto a smooth fan with a closed set of end-points. Note that S(X) is closed, it contains no nondegerate subcontinuum and thus it is 0-dimensional. Denote by v the vertex of X and choose a closed neighbourhood U of v such that diam(u) < ε and Bd (U) E(X) =. The required function is just shrinking U to a point. Therefore for the rest of the proof we may assume that E(X) is closed. Since X is a smooth fan with a closed set of end-points, we may describe X as a cone over the 0-dimensional set E(X). Thus every point of X can be represented as a pair (x, t), where x E(X) and t [0, 1] with the identification (x, 1) = (y, 1) = v for any x, y E(X). Since X in not locally connected, the derivative E(X) d is non-empty. Choose a point e E(X) d and represent the open subset E(X)\{e} of E(X) as the union E(X)\{e} = U n such that n=1 (1) U n is a closed and open subset of E(X), (2) diam (U n ) < ε, (3) U n U m = for n m. Identifying points (x, t) with (y, t), if x, y U n for some n {1, 2,... } 11

12 we get the required confluent (even open) ε-map of X onto the harmonic fan. The Lelek fan can be defined a smooth fan with a dense set of end-points. It has been constructed by Lelek in [16, 9, p. 314] and its uniqueness has been shown in [12]. Theorem Int F ({Lelek fan}) = {Lelek fan}, 2. Cl F ({Lelek fan}) = {Lelek fan}. Proof. The equation (1) follows from the fact that confluent image of the Lelek fan is the Lelek fan, see [12, Proposition 2, p. 32]. To see (2) assume that X Cl F ({Lelek fan}). We will show that X is a smooth fan with a dense set of end points, so it is homeomorphic to the Lelek fan by [12, Theorem, p. 31]. Really, X is a smooth fan by Theorem 4.3 (2), so assume on the contrary that the set of end points of X is not dense. Let x 0 X and ε > 0 be such that B(x 0, 2ε) contains no end point of X and let f : X Y be a confluent ε-map of X onto the Lelek fan Y. By [17, Lemma , p. 113] there is δ > 0 satisfying f 1 (B(f(x 0 ), δ)) B(x 0, ε). (1) Since the set of end points of the Lelek fan is dense, there is y E(Y ) B(f(x 0 ), δ). Because of E(Y ) f(e(x)) (see [4, Theorem 4.1 (10), p. 15]), there is e E(X) such that f(e) = y. On the other hand by (1) there is p B(x 0, ε) satisfying f(p) = y. Since p B(x 0, ε) and e / B(x 0, 2ε), we have d(p, e) > ε and f(p) = f(e) = y, so f is not an ε-mapping, a contradiction. We will call the countable union of copies C n of Cantor fans of diameter tending to 0 and having a common vertex the ω-cantor fan. Proposition 4.8. Any Kelley fan in which just end points do not contain isolated points is homeomorphic to the Cantor fan or to the ω-cantor fan. Proof. By [6, Corollary 4, 4 o ] X can be embedded into the ω-cantor fan in such a way that the end points of X are mapped into the end points of the ω-cantor fan. If the set E(X) of the end points of X is closed then E(X) is homeomorphic to the Cantor set, so X is homeomorphic to the Cantor fan. 12

13 Note that S(X) = E(X) {v} is closed by [7, Theorem 3 (c)]. If E(X) is not closed, then S(X) is a closed zero-dimensional set with no isolated points, so it is homeomorphic to the Cantor set. In this case one can show that X is homeomorphic to the ω-cantor fan. Theorem Int F ({Cantor fan}) =, 2. Cl F ({Cantor fan}) = {Cantor fan, ω-cantor fan}, 3. Int F ({ω-cantor fan}) =, 4. Cl F ({ω-cantor fan}) = {ω-cantor fan}. Proof. To see (1) and (3) observe that, for every ε > 0, there is a confluent ε-retraction of the Cantor fan or the ω-cantor fan onto an n-odd for some n. To see (2), let X Cl F ({Cantor fan}), by Theorem 4.4 (2) X is a Kelley fan, so it is a smooth fan, in other words it is a subcontinuum of the Cantor fan. To conclude the proof we need to show, by Proposition 4.8, that X does not have isolated end points. If p were an isolated end point, we could consider ε < dist (p, S(X)\{p}). If we take a confluent ε-map the image of the point p would be an isolated end point inside the Cantor fan, a contradiction. Therefore X is the Cantor fan or is the ω-cantor fan. Of course there are confluent (even monotone) ε-maps from the ω-cantor fan onto the Cantor fan. Thus Cl F ({Cantor fan}) = {Cantor fan, ω-cantor fan}. In order to prove (4) we observe that there are no confluent ε-maps from the Cantor fan onto the ω-cantor fan, and using the same argument as for the Cantor fan we get that Cl F ({ω-cantor fan}) = {ω-cantor fan}. Theorem Int F ({F ω }) =, 2. Cl F ({F ω }) = (K F) \ CF. Proof. The proof of (1) is similar to the proof of Theorem 4.9 (1). To prove Cl F ({F ω }) (K F) recall that F ω is a Kelley fan, so Cl F ({F ω }) Cl F (K F) = K F by Theorem 4.4 (2). On the other hand (see Theorem 4.5 (1)) Int F CF = CF and F ω / CF so no member of Cl F ({F ω }) is in CF. Thus Cl F ({F ω }) (K F) \ CF. In order to prove (K F) \ CF Cl F ({F ω }), let X (K F) \ CF and ε > 0. Recall that, by [6, Theorem 3, p. 171], there exists an embedding f of X 13

14 into the ω-cantor such that end points of X are mapped to end points of the ω-cantor fan. So, since X / CF, there are infinitely many copies C nk of the Cantor fan in the ω-cantor fan such that f(x) C nk {v}, where v is the vertex of f(x). On the other hand for each C nk, there is an open ε-retraction onto an n-od. Thus, the union of those retractions, restricted to f(x) is an open, thus confluent, ε-map from f(x) onto F ω. The proof of the following lemma is a consequence of the definition of the Hausdorff distance H d. Lemma If f : X Y is an ε-mapping, for some ε > 0, and A and B are closed subsets of X such that f(a) = f(b), then H d (A, B) < ε. In other words, the induced function 2 f : 2 X 2 Y is also an ε-mapping. Example There is a family {F λ : λ Λ} of cardinality c such that Int F ({F λ }) = {F λ }. Consequently, each {F λ } is an open set in (N, τ F ), and thus the density of the space (N, τ F ) is c. Figure 1: The fan F 14

15 Proof. Let F be the fan as pictured in Figure 1. Here we have L n = L n,0 L n,n, where L n,k for 0 k n is an arc, lim L n,0 = {p}, liml n,k = L n for k 1, L n,k L n,k+1 is a one-point set, L n,k L n,k = if n n or if k k > 1. For a given 0, 1-sequence λ, we define F λ = L {L n : λ(n) = 1}, i.e. we include L n in F λ if and only if the n-th place in λ is 1. In particular F = F 1,1,1,.... We will divide the proof into several steps. Let a m,n 1 and a m,n be end points of L m,n. Let d be a metric in F λ such that diam (L m,n ) = d(a m,n 1, a m,n ) = 1, for each m {1, 2,... } and n {0, 1, 2,..., m} with n m. Moreover we assume that the Hausdorff distance generated from d satisfies H d (L n,m, L m,n ) < 1 and that there are points 4 b m,n L m,n such that diam (b m,n a m,n ) = d(b m,n, a m,n ) = diam (a m,n b m,n+1 ) = d(a m,n, b m,n+1 ) = 1 for all m {1, 2,... } and n {0, 1,... m}. 2 Let ε be a number less than min{ 1, d(a 4 1,1, a 3,3 ), d(a 2,2, p)}. Then the images of a 1,1, a 2,2, a 3,3, and p under any ε-mapping are four different points. Step 1. Every confluent ε-mapping defined on F λ is monotone. Let us recall that the image of a fan under a confluent mapping is a fan or an arc (see [3, Theorem 12, p. 32]). By the choice of ε, the image of F λ under a confluent ε-mapping has a ramification point, so it is a fan. Recall also that confluent mappings between fans are monotone relative to the top (see [8, Theorem 4.1, p. 14]). Suppose that f defined on F λ is a confluent ε-mapping but it is not monotone. Then, by [8, Theorem 4.1, p. 14], there exist two end points a m,m and a n,n of F λ such that n < m and f(a m,m ) = f(a n,n ). Then the arc L n is the union L n = L n,0 a n,0 b n,1 b n,1 b n,2 b n,n 1 b n,n b n,n a n,n, of n + 2 arcs, each of diameter 1 or less. 2 Denote by c i, for i {1, 2,..., n}, a point of L m such that f(c i ) = f(b n,i ). Then, as previously we have L m = L m,0 a n,0 c 1 c 1 c 2 c n 1 c n c n a m,m. Thus L m is written as the union of n+2 arcs. Because m > n, one of the arcs has to have the diameter equal to 1. Since the images of the respective arcs are the same, there are two arcs A and B, A L m and B L n, satisfying diam (A) = 1, diam (B) = 1 and f(a) = f(b). This contradicts Lemma , and thus Step 1 is shown. 15

16 Step 2. Every image of F λ under a monotone ε-mapping is homeomorphic to F λ. Let f : F λ Y be a monotone ε-mapping. We will construct a homeomorphism h : Y F λ. To this end define a function ϕ from a closed subset of Y onto [0, 1] in the following way: let ϕ f(l) be any homeomorphism of f(l) onto [0, 1] with f(p) = 0; let ϕ(f(a m,2n )) = 0 and ϕ(f(a m,2n+1 )) = 1 for each m such that λ(m) = 1. Let ϕ be the extension of ϕ to Y. We aim to modify ϕ in such a way that the modification is monotone on each arc f(l m,n ), for m, n such that L m,n F λ. To this end define ϕ 1 (x) = max{ϕ (y) : y a m,n 1 x}, if n is even, ϕ 1 (x) = min{ϕ (y) : y a m,n 1 x} if n is odd, and ϕ 1 (x) = ϕ (x) for x L {L m,0 : λ(m) = 1}. Next, let ϕ 2 : Y [0, 1] be any map such that ϕ 2 (x) = ϕ 1 (x) = ϕ (x) for x L {L m,0 : λ(m) = 1} and ϕ 2 f(lm,n) is a homeomorphism satisfying dist (ϕ 2 f(lm,n), ϕ 1 f(lm,n)) < 1 m. In other words, we are changing the monotone map ϕ 1 f(lm,n) by a close homeomorphism ϕ 2 f(lm,n). Finally define h : Y F λ by the following conditions: 1. If y f(l), h(y) is the unique point of L with d(p, h(y)) = ϕ 2 (y). 2. h f(lm,0 ) is any homeomorphism of f(l m,0 ) into L m,0 satisfying h(f(p)) = p. 3. If y f(l m,n ), then h(y) is the unique point of L m,n satisfying d(a m,n 1, h(y)) = ϕ 2 (y), if n is odd and d(a m,n, h(y)) = ϕ 2 (y), if n is even. The reader may verify that h is the required homeomorphism. Step 3. To see that the family {F λ : λ Λ} has cardinality c observe that F λ is homeomorphic to F λ if and only if λ(n) λ (n) for finitely many n and (λ(n) λ (n)) = 0. n=1 First suppose that λ(n) λ (n) for finitely many n and n=1 (λ(n) λ (n)) = 0, then there is a number n 0 such that λ(n) = λ (n) for n n 0 and card {n : λ (n) = 1 and n < n 0 } = card {n : λ(n) = 1 and n < n 0 }. Let φ : {n : λ(n) = 1 and n < n 0 } {n : λ (n) = 1 and n < n 0 } be a bijection. Then one can construct a homeomorphism h : F λ F λ such that h Ln is the identity if n n 0 and h Ln : L n L φ(n) is a homeomorphism if n < n 0. Second assume we have a homeomorphism h : F λ F λ. Then h(l) = L, and, by continuity there is n 0 such that λ(n) = λ (n) for n n 0. Moreover h(l λn ) = L λ n if λ(n) = λ (n) = 1 and n n 0. Finally we have card {n : λ (n) = 1 and n < n 0 } = card {n : λ(n) = 1 and n < n 0 }. The proof is finished. 16

17 Problem Is it true that Cl F ({F λ }) = {F λ } for each 0,1-sequence λ? Problem Characterize fans X such that Int F ({X}) = Cl F ({X}) = {X}. We know that n-ods, for n 3 and the Lelek fan satisfy the last equation, and we conjecture that all fans F λ from Example 4.12 satisfy it. 5. Dendrites, dendroids and λ-dendroids In this section we will recall results about the following classes of continua: TR, D 0, SD, D, λd and TL. First, observe the following inclusions TR D 0 SD D λd TL. Second, notice that all of the classes are confluent invariants, and thus all of them are open in (N, τ F ). Precisely, we have the following theorem. Theorem Int F (TR) = TR, 2. Int F (D 0 ) = D 0, 3. Int F (SD) = SD, 4. Int F (D) = D, 5. Int F (λd) = λd, 6. Int F (TL) = TL. Proof. In all cases we will recall the results that the given class is invariant under confluent mappings. For (1), see [20, Corollary 13.43, p. 297]. The equations (2), (4), and (5) are consequences of the facts that confluent images of λ-dendroids are λ-dendroids, see [2, XIV, p. 217], because dendroids are arc-wise connected λ-dendroids and dendrites are locally connected λ- dendroids. The equations (3) and (6) follow from theorems by Maćkowiak and McLean in [18] and [19], respectively. Concerning the closures of the classes, we do not have any characterizations (except the class TL which is closed). Thus we gather only some properties of the closures of the classes. 17

18 Theorem Each element of Cl F (TR) can be represented as an inverse limit of trees with confluent bonding mappings, 2. Cl F (TR) = Cl F (D 0 ) AR(HU), 3. Every tree-like continuum can be embedded in a member of Cl F (TR), 4. Cl F (TR) AK. Proof. The equation (1) follows from [21, Theorem 3.2, p. 120]; the equations (2) and (3) were proved in [10, Theorem 3.6, p. 94, and Theorem 4.5, p. 99], respectively; the inclusion (4) follows (2) and [9, Corollary 3.7, p. 60]. Problem 5.3. Characterize the classes Cl F (TR), Cl F (SD), Cl F (D) and Cl F (λd). The following example is a counterexample to a hypothesis we had about Cl F (SD). Example 5.4. Cl F (TR) SD Cl F (SD). Figure 2: Dendroid X 1. Proof. We will construct the needed continuum X as an inverse limit of smooth dendroids with open bonding mappings. Let X 1 be the continuum pictured in Figure 2. We construct X 2 ; see Figure 3, as two copies of X 1 joint together by the lowest point in the picture of X 1 (the point marked by the star in the Figure 2). The map f 1 : X 2 X 1 is just identifying the two 18

19 Figure 3: Dendroid X 2. Figure 4: Dendroid X 3. 19

20 copies of X 1 onto X 1. Given X n we construct X n+1 as two copies of X n joint by the respective end point of X n, and f : X n+1 X n is, again, identifying the two copies of X n onto X n. The dendroid X 3 is pictured in Figure 4. Let X be the inverse limit lim{x n, f n }. Then natural projections from X onto X n are open (thus confluent) ε-mappings onto smooth dendroids X n, so X Cl F (SD). The inverse limit of the vertical segments is the simplest indecomposable continuum, so X is not a dendroid; X is not a Kelley continuum, because it admits a confluent mapping onto a non-kelley dendroid X 1, so X / Cl F (TR). Acknowledgement The authors would like to thank David Maya for his help in preparation of this article, especially in making the pictures. References [1] J. G. Anaya, F. Capulín, E. Castañeda-Alvarado, W. J. Charatonik and F. Orozco-Zitli, On representation spaces, Topology Appl. 164 (2014), [2] J. J. Charatonik, Confluent mappings and unicoherence of continua, Fund. Math., LVI (1964), [3] J. J. Charatonik, On fans, Dissertations Math. (Rozprawy Mat.) 54 (1967), [4] J. J. Charatonik, W. J. Charatonik and S. Miklos, Confluent mappings of fans, Dissertations Math. (Rozprawy Mat.) 301 (1990), [5] F. Capulín, R. Escobedo, F. Orozco-Zitli and I. Puga, On ε-properties, Selected Papers of the 2010 International Conference of Topology and its Applications, Nafpaktos, Greece, Technological Educational Institute of Messolonghi, (2010), [6] J. J. Charatonik, W. J. Charatonik, The property of Kelley for fans, Bull. Polish. Acad. Sci. Math. 36 (1988),

21 [7] J. J. Charatonik, W. J. Charatonik, Fans with the property of Kelley, Topology Appl. 29 (1988), [8] J. J. Charatonik, W. J. Charatonik and S. Miklos, Confluent mappings of fans, Dissertationes Math. (Rozprawy Mat.) 301 (1990), [9] J. J. Charatonik, W. J. Charatonik and J. R. Prajs, Arc property of Kelley and absolute retracts for hereditarily unicoherent continua, Colloq. Math., 97 (2003), [10] J. J. Charatonik, W. J. Charatonik and J. R. Prajs, Hereditarily unicoherent continua and their absolute retracts, Rocky Mountain J. Math., 34 (2004), [11] J. J. Charatonik, W. J. Charatonik and J. R. Prajs, Confluent mappings and arc Kelley Continua, Rocky Mountain J. Math., 38 (2008), [12] W. J. Charatonik, The Lelek fan is unique, Houston J. Math, 15 (1989), [13] W. J. Charatonik, M. Insall and J. R. Prajs, Connectedness of the representation space for continua, Topology Proc. 40 (2012), [14] H. Freudenthal, Entwicklungen von Räumen und ihren Gruppen, Compositio Math., 4(1937), [15] A. Illanes, A characterization of dendroids by the n-connectedness of the Whitney levels, Fund. Math., 140 (1992), [16] A. Lelek, On plain dendroids and their end-points in the classical sense, Fund. Math., 49 (1961), [17] S. Macías, Topics on continua, Chapman & Hall/CRC, Boca Raton, FL, [18] T. Maćkowiak, Confluent mappings and smothness of dendroids, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 21 (1973), [19] B. McLean, Confluent images of tree-like curves are tree-like, Duke Math. J., 39 (1972),

22 [20] S. B. Nadler, Jr., Continuum theory, Marcel Dekker, New York, [21] L.G. Oversteegen and J. R. Prajs, On confluently graph-like compacta, Fund. Math., 178 (2003), [22] S. A. Smith, When is one graph the weakly confluent image of another?, Topology Proc. 20 (1995), [23] R.H. Sorgenfrey, Concerning triodic continua, Amer. J. Math. 66 (1944),