HOUSTON JOURNAL OF
MATHEMATICS

Electronic Edition Vol. 26, No. 4, 2000

Editors: H. Amann (Z├╝rich), G. Auchmuty (Houston), D. Bao (Houston), H. Brezis (Paris), S. S. Chern (Berkeley), J. Damon (Chapel Hill), K. Davidson (Waterloo), C. Hagopian (Sacramento), R. M. Hardt (Rice), J. Hausen (Houston), J. A. Johnson (Houston), J. Nagata (Osaka), B. H. Neumann (Canberra), G. Pisier (College Station and Paris), S. W. Semmes (Rice)
Managing Editor: K. Kaiser (Houston)


This issue is dedicated to Professor Andrew Lelek in thanks for, and honor of, his twenty-five years of service as an editor of the Houston Journal of Mathematics. Professor Lelek was one of the founding editors in the establishment of the HJM in 1975. He served as the Managing Editor for four years, from 1981 through 1984. Under Professor Lelek's stewardship, the HJM has become known for its publications in topology.

After a careful review, editor Charles Hagopian has selected fifteen papers in different areas of topology for inclusion in this issue from about thirty accepted submissions. As such, these papers not only reflect the taste of Professor Lelek but constitute a lively snapshot of current topology.

For the editorial board,
Charles Hagopian
Klaus Kaiser

Contents

Harold R Bennett, Department of Mathematics, Texas Tech University, Lubbock, TX., U.S.A. (graddir@math.ttu.edu), Masami Hosobuchi, Department of Housing and Planning, Tokyo Kasei Gakuin University, Machida, Tokyo, JAPAN (mhsbc@kasei-gakuin.ac.jp) and David J. Lutzer, Department of Mathematics, College of William and Mary, Williamsburg, VA., U.S.A. (lutzer@math.wm.edu).
Weakly Perfect Generalized Ordered Spaces, pp. 609-627.
ABSTRACT. A space X is weakly perfect if each closed subset of X contains a dense subset that is a Gdelta-subset of X. This property was introduced by Kocinac and later studied by Heath. We provide three mechanisms for constructing ZFC examples of spaces that are weakly perfect but not perfect. Some of our examples are compact linearly ordered spaces, while others are types of Michael lines. Our constructions begin with special subsets of the usual unit interval, e.g., perfectly meager subsets. We conclude by giving a new and strictly internal topological characterization of perfectly meager subsets of [0,1], namely that a topological space X is homeomorphic to a perfectly meager subset of [0,1] if and only if X is a zero-dimensional separable metrizable space with the property that every subset A of X contains a countable set B that is dense in A and is a Gdelta-subset of X.

L. Block, University of Florida (block@math.ufl.edu) , J. Keesling, University of Florida (jek@math.ufl.edu) and V.V. Uspenskij, Ohio University (uspensk@math.ohiou.edu).
Inverse Limits which are the Pseudoarc , pp. 629-638.
ABSTRACT. Let Cs(I,I) denote the space of surjective continuous maps of the compact interval I to itself with the uniform topology. Given a map f in Cs(I,I), let (I,f) denote the inverse limit space obtained from the inverse sequence all of whose maps are f and all of whose spaces are I. We show that the set of f in Cs(I,I) such that (I,f) is homeomorphic to the pseudoarc is nowhere dense in Cs(I,I). Also, we show that if f is any continuous map of I to itself such that f has a periodic point of period two or larger, but f has no periodic point of odd period larger than one, then (I,f) is not homeomorphic to the pseudoarc. It follows that if f is any continuous map of I to itself with (I,f) the pseudoarc and with topological entropy positive, then the topological entropy of f is greater than log(2)/2.

Janusz J. Charatonik, Mathematical Institute, University of Wroclaw, pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland (jjc@hera.math.uni.wroc.pl).
On generalized rigidity , pp. 639-660.
ABSTRACT. Concepts of chaotic and of rigid spaces with respect to a given class of mappings are introduced and studied in the paper. A special attention is paid to the classes of open and of monotone mappings. The obtained results are applied to dendrites.

Alex Clark, Department of Mathematics, University of North Texas, Denton, TX 76203-1430 (alexc@unt.edu).
Solenoidalization and Denjoids, pp. 661-692.
ABSTRACT. We describe a method (solenoidalization) of obtaining flows on kappa --solenoids from a given flow on a kappa --torus. When we apply this process to the Denjoy flows on T2 we obtain flows whose minimal sets we call denjoids. We give a topological classification of these indecomposable, one-dimensional continua.

J. F. Davis and Sam B. Nadler Jr., Department of Mathematics, West Virginia University, P.O. Box 6310, Morgantown, WV 26506-6310 (nadler@math.wvu.edu).
Hereditarily Weakly Confluent Mappings Onto S1 , pp. 693--720.
ABSTRACT. Results are obtained about the existence and behavior of hereditarily weakly confluent maps of continua onto the unit circle S1. A simple and useful necessary and sufficient condition is given for a map of a continuum, X, onto S1 to be hereditarily weakly confluent (HWC). It is shown that when X is arcwise connected, an HWC map of X onto S1 is monotone with arcwise connected fibers. A number of theorems about HWC irreducible maps of X onto S1 are proved; for example, such maps are monotone with nowhere dense fibers, and a complete determination of the structure of X is obtained when X admits an HWC irreducible map onto S1 and X is arcwise connected. Among other results, the arcwise connected semi-locally-connected continua that admit an HWC map onto S1 are completely determined, and it is shown how the map must be defined.

Valentin Gutev, School of Mathematical and Statistical Sciences, Faculty of Science, University of Natal, King George V Avenue, Durban 4041, South Africa.
An Exponential Mapping Over Set-Valued Mappings, pp. 721-739.
ABSTRACT. The paper presents an approach to ``selection homotopy extension'' properties of set-valued mappings showing that they become equivalent to usual selection extension properties of exponential set-valued mappings associated to them. As a result, several ``controlled'' homotopy extension theorems are obtained like consequences of ordinary selection theorems. Also, involving set-valued mappings, a simple proof of the Borsuk homotopy extension theorem is given.

K.T. Hallenbeck, Department of Mathematics, Widener University, Chester, Pa 19013.
Estimates of Spans of a Simple Closed Curve Involving Mesh, pp. 741-745.
ABSTRACT. We show that the dual effectively monotone span of a simple closed curve X in the plane does not exceed the infimum of the set of positive numbers m such that a chain with mesh m covers X. We also include a short direct proof of a known inequality sigma0 (0) <= epsilon (X) where X is a continuum.

David Handel, Department of Mathematics, Wayne State University, Detroit, Michigan 48202, USA (handel@math.wayne.edu).
Some Homotopy Properties of Spaces of Finite Subsets of Topological Spaces, pp. 747-764.
ABSTRACT. For X a non-empty topological space and k a positive integer, we denote by Sub(X,k) the set of non-empty subsets of X having cardinality less than or equal to k, suitably topologized. The Sub(- ,k) are homotopy functors and their properties are studied. We prove that if X is Hausdorff and path-connected, then for all integers k greater than or equal to 1, the inclusion maps from Sub(X,k) into Sub(X,2k+1) induce the trivial homomorphisms in all homotopy groups. In the direction of non-triviality, we prove that if X is a non-empty closed manifold of dimension at least 2, then for each positive integer k, Sub(X,k) is homologically non-trivial.

Morris W. Hirsch, Department of Mathematics, University of California, Berkeley, CA 94720-3840 (hirsch@math.berkeley.edu) .
Topology of Fixed Point Sets of Surface Homeomorphisms , pp. 765-789.
ABSTRACT. This paper investigates the topology of the fixed point set F of an orientation preserving homeomorphism of a connected surface M, under the assumptions that M has finitely generated homology, F is compact and nonempty with finitely many components, and no complementary component of F is an open cell. This last condition holds if area is preserved, or nonwandering points are dense, or there is a nowhere dense global attractor. The main conclusion is that the Euler characteristic of F for Cech cohomology is finite and no smaller than the Euler characteristic of M. Applications are made to attractors, analytic homeomorphisms, homoclinic points, prime power iterates, and commuting homeomorphisms.

Tetsuya Hosaka, Institute of Mathematics, University of Tsukuba, Tsukuba, 305-8571, Japan, (thosaka@math.tsukuba.ac.jp), and Katsuya Yokoi, Department of Mathematics, Interdisciplinary faculty of Science and Engineering, Shimane University, Matsue, 690-8504, Japan (yokoi@math.shimane-u.ac.jp).
The Boundary and the Virtual Cohomological Dimension of Coxeter Groups, pp. 791-805.
ABSTRACT. This paper consists of three parts:
1) We give some properties about the virtual cohomological dimension (vcd, for short) of Coxeter groups over principal ideal domains.
2) For a right-angled Coxeter group with n-vcd over a PID R, we construct a sequence of parabolic subgroups with i-vcd over R for i less than or equal to n.
3) We show that a parabolic subgroup of a right-angled Coxeter group is of finite index if and only if their boundaries coincide.

Ondrej F.K. Kalenda, Department of Math. Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovska 83, 186 75 Praha 8, Czech Republic} (kalenda@karlin.mff.cuni.cz).
On Double-Derived Sets in Topological Spaces , pp. 807-809.
ABSTRACT. We characterize topological spaces which have a subset with non-closed double-derived set. As a corollary we obtain that the double-derived set of an arbitrary subset of a T0 topological space is closed. This answers in the negative a question asked by A.Lelek in Houston Problem Book (1995).

Takuo Miwa, Department of Mathematics, Shimane University, Matsue 690-8504, Japan (miwa@math.shimane-u.ac.jp). On Fibrewise Retraction and Extension , pp. 811-831.
ABSTRACT. We study fibrewise retracts and extensions. We introduce notions of absolute (nbd) retracts (or extensor) over a base space B relative to a fibrewise class Q, and a notion of fibrewise adjunction spaces. We study the relations of fibrewise ANR and ANE, and fibrewise contractibility and fibrewise ANE.

Robert Pierce, Dr. Robert Pierce, 3260 Schneider Rd. #106, Toledo Ohio 43614 (Bobscram@aol.com).
Special Unions of Unicoherent Continua , pp. 833-868.
ABSTRACT. It is proved that a Hausdorff continuum is unicoherent if it is the union of two unicoherent continua whose intersection is connected and locally connected.

Elzbieta Pol and Roman Pol, Institute of Mathematics, University of Warzaw, Banacha 2, 02-097 Warzaw, Poland (pol@mimuw.edu.pl) .
On the Krasinkiewicz - Minc Theorem concerning Countable Fans , pp. 869-876.
ABSTRACT. A strengthening of a remarkable theorem of Krasinkiewicz and Minc is discussed to the effect that there are planar fans Dalpha, alpha < omega 1, such that if X is completely metrizable separable and each Dalpha is a continuous (homeomorphic) image of a continuum in X, then so is every chainable continuum. We shall also give an analogous strengthening of a theorem of Mackowiak concerning hereditarily decomposable chainable continua.

Yun Ziqiu, Department of Mathematics Suzhou University Suzhou, Jiangsu People's Republic of China (yunziqiu@public1.sz.js.cn) and Heikki J.K. Junnila, Department of Mathematics University of Helsinki Helsinki Finland (heikki.junnila@helsinki.fi).
On a Special Metric , pp. 877-882.
ABSTRACT. In this note, we prove that whenever d is a compatible metric for a sufficiently large hedgehog space J, there exist a positive number r and a point x of the space J such that the family {B(y,r): d(y,x)<r} contains uncountably many distinct sets. This result provides a negative answer to a question raised by J. Nagata. We also give positive answers to the same question under some extra conditions.