Hostname: page-component-5d84bcc8dc-29vhl Total loading time: 0 Render date: 2026-09-04T03:13:29.425Z Has data issue: false hasContentIssue false
  • English
  • Français

REFLEXIVITY INDEX OF THE PRODUCT OF SOME TOPOLOGICAL SPACES AND LATTICES

Part of: Lattices Linear function spaces and their duals

Published online by Cambridge University Press:  27 March 2023

BINGZHANG MA Open the ORCID record for BINGZHANG MA [Opens in a new window]
BINGZHANG MA*
Affiliation:
School of Mathematics, East China University of Science and Technology, Shanghai, PR China
*
e-mail: bingzhangmath@foxmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce a technique that is helpful in evaluating the reflexivity index of several classes of topological spaces and lattices. The main results are related to products: we give a sufficient condition for the product of a topological space and a nest of balls to have low reflexivity index and determine the reflexivity index of all compact connected 2-manifolds.

Keywords

closed set latticereflexivereflexivityreflexivity index

MSC classification

Primary: 46E05: Lattices of continuous, differentiable or analytic functions
Secondary: 06B30: Topological lattices, order topologies

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 109 , Issue 2 , April 2024 , pp. 376 - 387
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

This research was supported by the National Natural Science Foundation of China (Grant No. 11871021).

References

Alexander, B., Lex, O. and Tymchatyn, E. D., ‘On minimal maps of 2-manifolds’, Ergodic Theory Dynam. Systems 25 (2005), 41–57.Google Scholar
Davidson, K. R., Nest Algebras. Triangular Forms for Operator Algebras on Hilbert Space (Longman, New York, 1988).Google Scholar
Gottschalk, W. H., ‘Orbit-closure decompositions and almost periodic properties’, Bull. Amer. Math. Soc. (N.S.) 50 (1944), 915–919.CrossRefGoogle Scholar
Halmos, P. R., ‘Reflexive lattices of subspaces’, J. Lond. Math. Soc. (2) 4 (1971), 257–263.CrossRefGoogle Scholar
Harrison, K. J. and Ward, J. A., ‘Reflexive nests of finite subsets of a Banach space’, J. Math. Anal. Appl. 420 (2014), 1468–1477.CrossRefGoogle Scholar
Harrison, K. J. and Ward, J. A., ‘The reflexivity index of a finite distributive lattice of subspaces’, Linear Algebra Appl. 455 (2014), 73–81.CrossRefGoogle Scholar
Harrison, K. J. and Ward, J. A., ‘The reflexivity index of a lattice of sets’, J. Aust. Math. Soc. 97 (2014), 237–250.CrossRefGoogle Scholar
Hatcher, A., Algebraic Topology (Cambridge University Press, Cambridge, 2002).Google Scholar
Kronecker, L., N $\ddot{a}$ herungsweise ganzzahlige Aufl $\ddot{o}$ sung linearer Gleichungen (Chelsea, New York, 1968).Google Scholar
Longstaff, W. E., ‘Reflexive index of a family of subspaces’, Bull. Aust. Math. Soc. 90 (2014), 134–140.CrossRefGoogle Scholar
Ma, B. and Harrison, K. J., ‘Reflexivity index and irrational rotations’, Bull. Aust. Math. Soc. 104 (2021), 493–505.CrossRefGoogle Scholar
Moise, E. E., Geometric Topology in Dimensions 2 and 3 (Springer-Verlag, New York, 1977).CrossRefGoogle Scholar
Yang, Z. Q. and Zhao, D. S., ‘On reflexive closed set lattices’, Comment. Math. Univ. Carolin. 51 (2010), 143–154.Google Scholar
Zhao, D. S., ‘Reflexive index of a family of sets’, Kyungpook Math. J. 54 (2014), 263–269.CrossRefGoogle Scholar