The Planck boundary within the hyperspace of the circle of pseudo-arcs. (English) Zbl 1538.54092
“The main goal of this paper is to reveal interesting geometric structures emerging from the study of hyperspaces of decomposable, non-locally connected homogeneous continua, in particular the hyperspaces of the circle of pseudo-arcs and the Menger curve of pseudo-arcs.”
The authors are mainly interested in \(C(X)\), the hyperspace of subcontinua of such \(X\).
The first part of the paper presents an expository review of some of the history of the developments involving pseudo-arcs, pseudo-circles, Whitney maps, and related topics. In Section 4 there are definitions of filament and ample subcontinua of a homogeneous continuum \(X\). These might be denoted \(\mathcal{F}(X)\) and \(\mathcal{A}(X)\), the former being “thin,” the latter being “thick.” If \(X\) is a decomposable nowhere locally connected homogeneous continuum, then the pair \(\{\mathcal{F}(X),\mathcal{A}(X)\}\) is a decomposition of \(C(X)\). This and other facts can be found in Theorem 4.1. If \(\Psi\) is a circle of pseudo-arcs, then by Definition 7.1, the Planck boundary of \(C(\Psi)\) is denoted, \(\mathcal{P}(\Psi)=\mathcal{A}(\Psi)\cap\overline{\mathcal{F}}(\Psi)\).
There is a continuous decomposition \(\mathcal{D}(\Psi)=\{P_\alpha\,\vert \,\alpha\in[0,2\pi)\}\) of \(\Psi\) into terminal pseudo-arcs \(P_\alpha\) (see page 459) whose quotient space is \(S^1\).
Theorem 7.1. The Planck boundary of \(C(\Psi)\) is a continuum in \(C(\Psi)\) consisting of the minimal ample subcontinua of \(\Psi\), which are the maximal terminal pseudo-arcs from the decomposition \(\mathcal{D}(\Psi)\).
The authors find a particular circle of pseudo-arcs (page 463) which they denote \(\Psi_0\) by employing a certain continuous decomposition of \(S^2\) and putting \(\Psi_0\) equal the inverse image of \(S^1\subset S^2\) under the quotient map. Using the Whitney metric, they find a specific \(l>0\) which they call the Planck constant of \(\Psi_0\). Theorem 7.2, which we shall not state, involves \(C(\Psi_0)\) which is to be endowed with the Whitney metric \(d_\mu\) as defined in Section 5.2. There are six statements in the conclusion of this theorem; we mention the first: (a) The length structure defined by the order-arcs has a nonnegative metric curvature.
In the final section, Section 8, there is a statement to the effect that there exists a Menger curve of pseudo-arcs. For this the authors state that a Planck boundary can be defined, and this happens to be a Menger curve. Other remarks about the hyperspace of subcontinua of the Menger curve are made.
The authors are mainly interested in \(C(X)\), the hyperspace of subcontinua of such \(X\).
The first part of the paper presents an expository review of some of the history of the developments involving pseudo-arcs, pseudo-circles, Whitney maps, and related topics. In Section 4 there are definitions of filament and ample subcontinua of a homogeneous continuum \(X\). These might be denoted \(\mathcal{F}(X)\) and \(\mathcal{A}(X)\), the former being “thin,” the latter being “thick.” If \(X\) is a decomposable nowhere locally connected homogeneous continuum, then the pair \(\{\mathcal{F}(X),\mathcal{A}(X)\}\) is a decomposition of \(C(X)\). This and other facts can be found in Theorem 4.1. If \(\Psi\) is a circle of pseudo-arcs, then by Definition 7.1, the Planck boundary of \(C(\Psi)\) is denoted, \(\mathcal{P}(\Psi)=\mathcal{A}(\Psi)\cap\overline{\mathcal{F}}(\Psi)\).
There is a continuous decomposition \(\mathcal{D}(\Psi)=\{P_\alpha\,\vert \,\alpha\in[0,2\pi)\}\) of \(\Psi\) into terminal pseudo-arcs \(P_\alpha\) (see page 459) whose quotient space is \(S^1\).
Theorem 7.1. The Planck boundary of \(C(\Psi)\) is a continuum in \(C(\Psi)\) consisting of the minimal ample subcontinua of \(\Psi\), which are the maximal terminal pseudo-arcs from the decomposition \(\mathcal{D}(\Psi)\).
The authors find a particular circle of pseudo-arcs (page 463) which they denote \(\Psi_0\) by employing a certain continuous decomposition of \(S^2\) and putting \(\Psi_0\) equal the inverse image of \(S^1\subset S^2\) under the quotient map. Using the Whitney metric, they find a specific \(l>0\) which they call the Planck constant of \(\Psi_0\). Theorem 7.2, which we shall not state, involves \(C(\Psi_0)\) which is to be endowed with the Whitney metric \(d_\mu\) as defined in Section 5.2. There are six statements in the conclusion of this theorem; we mention the first: (a) The length structure defined by the order-arcs has a nonnegative metric curvature.
In the final section, Section 8, there is a statement to the effect that there exists a Menger curve of pseudo-arcs. For this the authors state that a Planck boundary can be defined, and this happens to be a Menger curve. Other remarks about the hyperspace of subcontinua of the Menger curve are made.
Reviewer: Leonard R. Rubin (Norman)
MSC:
| 54F16 | Hyperspaces of continua |
| 51F99 | Metric geometry |
| 00A06 | Mathematics for nonmathematicians (engineering, social sciences, etc.) |
| 00A79 | Physics |
| 81-10 | Mathematical modeling or simulation for problems pertaining to quantum theory |
References:
| [1] | R. D. Anderson, One-dimensional continuous curves and homogeneity theorem, Ann. of Math., 68 (1958), 1-16. · Zbl 0083.17608 |
| [2] | R. H. Bing, A homogeneous indecomposable plane continuum, Duke Math. J., 15 (1948), 729-742. · Zbl 0035.39103 |
| [3] | R. H. Bing, Concerning hereditarily indecomposable continua, Pacific J. Math., 1 (1951), 43-51. · Zbl 0043.16803 |
| [4] | R. H. Bing and F. B. Jones, Another homogeneous plane continuum, Trans. Amer. Math. Soc., 90 (1959), 171-192. · Zbl 0084.18903 |
| [5] | D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, Graduate Studies in Mathematics, vol. 33, AMS (Providence, RI, 2001). · Zbl 0981.51016 |
| [6] | W. J. Charatonik, A metric on hyperspaces defined by Whitney maps, Proc. Amer. Math. Soc., 94 (1985), 535-538 · Zbl 0567.54007 |
| [7] | C. L. Hagopian, M. M. Marsh and J. R. Prajs, Folders of continua, Topology Proc., 55 (2020), 13-33. · Zbl 1441.54021 |
| [8] | C. L. Hagopian and J. T. Rogers, A classification of homogeneous circle-like continua, Houston J. Math., 3 (1977), 471-474. · Zbl 0397.54040 |
| [9] | L. C. Hoehn and L. G. Oversteegen, A complete classification of homogeneous plane continua, Acta Math., 216 (2016), 177-216. · Zbl 1361.54020 |
| [10] | A. Illanes, Pseudo-homotopies of the pseudo-arc, Comment. Math. Univ. Carolin., 53 (2012), 629-635. · Zbl 1274.54097 |
| [11] | A. Illanes and S. B. Nadler, Hyperspaces: Fundamentals and Recent Advances, Marcel Dekker, Inc. (New York-Basel, 1999). · Zbl 0933.54009 |
| [12] | F. B. Jones, On a certain type of homogeneous continua, Proc. Amer. Math. Soc., 6 (1955), 735-740. · Zbl 0067.40506 |
| [13] | B. Knaster, Un continu dont tout sous-continu est ind´ecomposable, Fund. Math., 3 (1922), 247-286. · JFM 48.0212.01 |
| [14] | W. Lewis, Homogeneous circle-like continua, Proc. Amer. Math. Soc., 89 (1983), 163- 168. · Zbl 0527.54031 |
| [15] | W. Lewis, Continuous curves of pseudo-arcs, Houston J. Math., 11 (1985), 91-99. · Zbl 0577.54039 |
| [16] | W. Lewis, Another characterization of the pseudo-arc, Topology Proc., 23 (1998), 235- 244. · Zbl 0945.54028 |
| [17] | W. Lewis, and J. J. Walsh, A continuous decomposition of the plane into pseudo-arcs, Houston J. Math., 4 (1978), 209-222. · Zbl 0393.54007 |
| [18] | E. E. Moise, An indecomposable plane continuum which is homeomorphic to each of its nondegenerate subcontinua, Trans. Amer. Math. Soc., 63 (1948), 581-594. · Zbl 0031.41801 |
| [19] | S. B. Nadler, Hyperspaces of Sets: A Text with Research Questions, Mercel Dekker, Inc. (New York, 1978). · Zbl 0432.54007 |
| [20] | J. R. Prajs, Continuous decompositions of Peano plane continua into pseudo-arcs, Fund. Math., 158 (1998), 23-40. · Zbl 0929.54023 |
| [21] | J. R. Prajs, A continuous circle of pseudo-arcs filling up the annulus, Trans. Amer. Math. Soc., 352 (2000), 1743-1757. · Zbl 0936.54019 |
| [22] | J. R. Prajs, A continuous decomposition of the Menger curve into pseudo-arcs, Proc. Amer. Math. Soc., 128 (2000), 2487-2491. · Zbl 0979.54011 |
| [23] | J. R. Prajs, Continuous pseudo-hairy spaces and continuous pseudo-fans, Fund. Math., 171 (2002), 101-116. · Zbl 0986.54047 |
| [24] | J. R. Prajs and K. Whittington, Filament sets and homogeneous continua, Topology Appl., 154 (2007), 1581-1591. · Zbl 1119.54025 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
