Abstract
A partial order on a set induces a natural topology on this set, and special properties of the partial order influence this topology significantly. These aspects lead to new and interesting fixed point theorems. The interconnections among partial order, topology and fixed point property were systematically investigated by Wallace [11], Ward [12] and Manka [6]. This chapter highlights these contributions to fixed point theory and supplements the theorems detailed in the preceding chapter.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Birkhoff, G.: Lattice Theory. American Mathematical Society, Providence (1948)
Borsuk, K.: A theorem on fixed points. Bull. Acad. Pol. Sci. 2, 17–20 (1954)
Connell, E.H.: Properties of fixed point spaces. Proc. Am. Math. Soc. 10, 974–979 (1959)
Klee, V.: An example related to the fixed point property. Niew arch. voor Wiskunde 8, 81–82 (1960)
Manka, R.: Association and fixed points. Fundam. Math. 91, 105–121 (1976)
Manka, R.: Connection between set theory and the fixed point property. Colloq. Math. 43, 177–184 (1987)
Manka, R.: On fixed point theorems for multifunctions in dendroids. Colloq. Math. 42, 185–192 (1987)
Nadler, S.B.: The fixed point property for continua. Apartaciones Matematicas 30 (2005)
Schweigert, G.E.: Fixed elements and periodic types for homeomorphisms on S.L.C continua. Am. J. Math. 66, 229–244 (1944)
Wallace, A.D.: Monotone transformations. Duke Math. J. 9, 487–506 (1943)
Wallace, A.D.: A fixed point theorem. Bull. Am. Math. Soc. 51, 413–416 (1945)
Ward Jr., L.E.: Partially ordered topological spaces. Proc. Am. Math. Soc. 5, 144–161 (1954)
Ward Jr., L.E.: A characterization of the fixed point property for a class of set-valued mappings. Fundam. Math. 50, 159–164 (1961)
Ward Jr., L.E.: Monotone surjections having more than one fixed point. Rocky Mt. J. Math. 4, 95–106 (1974)
Whyburn, G.T.: Analytic Topology. American Mathematical Society, Providence (1942)
Wilder, R.L.: Topology of Manifolds. American Mathematical Society, Providence (1949)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2018 Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Subrahmanyam, P.V. (2018). Partially Ordered Topological Spaces and Fixed Points. In: Elementary Fixed Point Theorems. Forum for Interdisciplinary Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-13-3158-9_4
Download citation
DOI: https://doi.org/10.1007/978-981-13-3158-9_4
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-13-3157-2
Online ISBN: 978-981-13-3158-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)