Hyperspaces which are cones Alejandro Illanes Universidad Nacional - PowerPoint PPT Presentation

Hyperspaces which are cones Alejandro Illanes Universidad Nacional Autonóma de México Workshop in Continuum Theory and Dynamical Systems, Nipissing University, May 2018

For a continuum X, we consider 2 X = { A  X : A is closed and A   }, C(X) = { A  2 X : A is connected }, C n (X) = { A  2 X : A has at most n components }, F n (X) = { A  2 X : A has at most n points }, C(X) = C 1 (X) and F 1 (X) = { {p} : p  X }.

C ([0,1]) = { [a,b] : 0 ≤ a ≤ b ≤ 1 } ~ { (a,b)  R 2 : 0 ≤ a ≤ b ≤ 1 }.

C(S 1 )

C p (X) = { A  C(X) : p  A }

C(T)

v B A { (x,0) : x in X } { {x} ; x in X } = F1(X)

Let X be a finite-dimensional continuum. --X has the cone=hyperspace property if there is a homeomorphism h : C(X) → Cone(X) such that (a) h(X) = Vertex, (b) h({x}) = (x,0) for each x є X. --C(X) is homeomorphic to Cone(X). --C(X) is homeomorphic to Cone(Z) for some continuum Z.

1. When X has the cone = hyperspace property? (C(X) finite-dimensional)

A continuum X is decomposable if X = A  B, where A and B are proper subcontinua. It is indecomposable if it is not decomposable and X is hereditarily decomposable if each nondegenerate subcontinuum of X is decomposable.

Theorem. (J. T. Rogers, Jr., 1972) If X is a finite-dimensional continuum with the cone = hyperspace property, then X is an arc a simple closed curve or an indescomposable continuum such that each nondegenerate subcontinuum of X is an arc.

Definition. Let A be a proper subcontinuum of a continuum X. Then X is of type N at A if there are sequences of continua A n , B n , C n and D n ; there are points p  q in A and there are sequences of points p n and q n such that A n → A, B n → A, C n → A, D n → A, p n → p, q n → q, and A n  B n = {p n } and A n  B n = {q n } for all n.

Theorem. (A. Illanes, 2001) Let X be a finite-dimensional continuum. If X has the cone = hyperspace property, then X is not of type N at any of its proper subcontinua. Problem. Suppose that X is an indecomposable finite-dimensional continuum such that every nondegenerate proper subcontinuum of X is an arc and X is not of type N at any of its proper subcontinuum. Then does X have the cone = hyperspace property?

2. When C(X) is homeomorphic to Cone(X)? (X finite-dimensional)

Theorem. (Nadler, 1977 & Rogers, 1973). Hereditarily decomposable continua such that C(X) is homeomorphic to cone(X)

3. When C(X) is homeomorphic to Cone(Z)? (Z finite-dimensional)

Nadler, 1978. “I can prove that if X is a locally connected continuum for which its hyperspace C(X) has finite dimensión and is homeomorphic to the cone of a continuum Z, then X and Z must be arcs or circles”

C(T)

Theorem. (S. Macías, 1997) Let X be a locally connected continuum. Then C(X) is homeomorphic to Cone(Z) for some finite-dimensional continuum Z if and only if X is: (a) an arc, (b) a simple closed curve or, (c) an m-od (for some m).

Theorem. (M. de J. López, 2002) Let X be a finite dimensional continuum for which there is a continuum Z and a homeomorphism h : C(X) → Cone(Z). If Y є C(X) and h(Y) = vertex, then (a) Y has the cone = hyperspace, (b) X \ Y is locally connected, (c) X \ Y has a finite number of components, (d) each component of X \ Y is either homeomorphic to [0,1) or to the real line R , (e) if some component of X \ Y is homeomorphic to R , then X \ Y = R , (f) If a subcontinuum A of X does not contain Y, then A is an arc or a singleton.

M. De J. López and A. Illanes, 2002 Classification of the continua X for which C(X) is homeomorphic to the Cone(Z), where Z is a finite-dimensional continuum. Case X is hereditarily decomposable.

Theorem. (M. de J. López, 2002) Let X be a finite dimensional continuum for which there is a continuum Z and a homeomorphism h : C(X) → Cone(Z). If Y є C(X) and h(Y) = vertex, then (a) Y has the cone = hyperspace, (b) X \ Y is locally connected, (c) X \ Y has a finite number of components, (d) each component of X \ Y is either homeomorphic to [0,1) or to the real line R , (e) if some component of X \ Y is homeomorphic to R , then X \ Y = R , (f) If a subcontinuum A of X does not contain Y, then A is an arc or a singleton.

X If X = Cone(Y), then H (X) = Cone({ A  H (X) : A intersects the base }).

Theorem. (M. de J. López, 2002) Let X be a finite dimensional continuum for which there is a continuum Z and a homeomorphism h : C(X) → Cone(Z). If Y є C(X) and h(Y) = vertex, then (a) Y has the cone = hyperspace, (b) X \ Y is locally connected, (c) X \ Y has a finite number of components, (d) each component of X \ Y is either homeomorphic to [0,1) or to the real line R , (e) if some component of X \ Y is homeomorphic to R , then X \ Y = R , (f) If a subcontinuum A of X does not contain Y, then A is an arc or a singleton.

4. When C n (X) is homeomorphic to Cone(Z)? (Z finite-dimensional) C n (X) = { A  2 X : A has at most n components }.

Example (R. Schori, 2002) . C 2 ([0,1]) ≈ [0,1] 4 . Example (A. Illanes, 2004) . C 2 (S 1 ) ≈ The cone over a solid torus.

Theorem (V. Martínez de la Vega, 2006) . Let X be a finite graph such that X is not a simple closed curve. Then TFAE: (a) X is an arc or a simple m-od, (b) C n (X) is a cone for some n, (c) C n (X) is a cone for all n. Question. Is C n (S 1 ) a cone for all (for some) n > 2?

Question. Is C 2 (sin(1/x)-continuum) a cone? Question. Is C 2 (a circle with a spiral) a cone? Question. Is C 2 (Solenoid) a cone? Question. Is C 2 (Buckethandle) a cone?

A free arc in a continuum X is an arc J in X joining points p and q such that J - {p,q} is open in X.

A locally connected continuum X is almost meshed if the union of the free arcs is dense in X.

Theorem (A. Illanes, V. Martínez de la Vega & D. Michalik, 2018) . Let X be an almost meshed continuum. Then TFAE: (a) X is an arc or a simple m-od, (b) X is a cone, (c) C n (X) is a cone for some n, (d) C n (X) is a cone for all n.

A continuum is a dendroid if it is arcwise connected and hereditarily unicoherent (the intersection of every two continua is connected). A fan is a dendroid with exactly one ramification point.

Theorem (S. Macías & S. B. Nadler, Jr., 2002). Let X be a fan. Then C(X) is a cone if and only if X is a cone. Theorem (A. Illanes, V. Martínez de la Vega & D. Michalik, 2018). Let X be a fan and n >2. Then C n (X) is a cone if and only if X is a cone. Question. Does the theorem above hold for n = 2?

5. When F n (X) is homeomorphic to Cone(Z)? (Z finite-dimensional) F n (X) = { A  2 X : A has at most n points }.

Theorem (E. Castañeda, 2004) . Let X be a finite graph. Then F 2 (X) is a cone if and only if X is an arc or X is an m-od for some m. Theorem (A. Illanes, V. Martínez de la Vega & D. Michalik, 2018) . Let X be a locally connected curve. Then TFAE. (a) X is an m-od for some m, (b) X is a cone, (c) F n (X) is a cone for some n > 1, (d) F n (X) is a cone for all n > 1.

Theorem (A. Illanes & V. Martínez de la Vega, 2017) . Let X be a fan. Then TFAE. (a) X is a cone, (b) F n (X) is a cone for some n > 1, (c) F n (X) is a cone for all n > 1.

Questions (A. Illanes & V. Martínez de la Vega) . (1) Suppose that X is a continuum such that for some n > 1, F n (X) is a cone, must X be a cone?, of special interest are the cases when X is finite- dimensional or X is 1-dimensional. (2) Suppose that X is a dendroid such that, for some n >1, F n (X) is a cone, must X be a cone? (3) Suppose that X is a 1-dimensional continuum and F n (X) is contractible for some n > 2, does this imply that X is a dendroid?

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