Fixed point sets of metric and nonmetric spaces

Authors:
John R. Martin and William Weiss

Journal:
Trans. Amer. Math. Soc. **284** (1984), 337-353

MSC:
Primary 54H25; Secondary 03E35, 54A35

DOI:
https://doi.org/10.1090/S0002-9947-1984-0742428-1

MathSciNet review:
742428

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A space is said to have the complete invariance property CIP if every nonempty closed subset of is the fixed point set of some self-mapping of . It is shown that connected subgroups of the plane and compact groups need not have CIP, and CIP need not be preserved by self-products of Peano continua, nonmetric manifolds or 0-dimensional spaces. Sufficient conditions are given for an infinite product of spaces to have CIP. In particular, an uncountable product of real lines, circles or two-point spaces has CIP. Examples are given which contrast the behavior of CIP in the nonmetric and metric cases, and examples of spaces are given where the existence of CIP is neither provable nor refutable with the usual axioms of set theory.

**[1]**Jon Barwise,*Monotone quantifiers and admissible sets*, Generalized recursion theory, II (Proc. Second Sympos., Univ. Oslo, Oslo, 1977) Stud. Logic Foundations Math., vol. 94, North-Holland, Amsterdam-New York, 1978, pp. 1–38. MR**516928****[2]**H. Cook,*Continua which admit only the identity mapping onto non-degenerate subcontinua*, Fund. Math.**60**(1967), 241–249. MR**0220249****[3]**Keith J. Devlin,*Fundamentals of contemporary set theory*, Springer-Verlag, New York-Heidelberg, 1979. Universitext. MR**541746****[4]**Ryszard Engelking,*Topologia ogólna*, Państwowe Wydawnictwo Naukowe, Warsaw, 1975 (Polish). Biblioteka Matematyczna. Tom 47. [Mathematics Library. Vol. 47]. MR**0500779**

Ryszard Engelking,*General topology*, PWN—Polish Scientific Publishers, Warsaw, 1977. Translated from the Polish by the author; Monografie Matematyczne, Tom 60. [Mathematical Monographs, Vol. 60]. MR**0500780****[5]**Bo Ju Jiang and Helga Schirmer,*Fixed point sets of continuous self-maps on polyhedra*, Fixed point theory (Sherbrooke, Que., 1980) Lecture Notes in Math., vol. 886, Springer, Berlin-New York, 1981, pp. 171–177. MR**643006****[6]**F. B. Jones,*Connected and disconnected plane sets and the functional equation 𝑓(𝑥)+𝑓(𝑦)=𝑓(𝑥+𝑦)*, Bull. Amer. Math. Soc.**48**(1942), 115–120. MR**0005906**, https://doi.org/10.1090/S0002-9904-1942-07615-4**[7]**I. Juhász and William Weiss,*Martin’s axiom and normality*, General Topology Appl.**9**(1978), no. 3, 263–274. MR**510908****[8]**Kenneth Kunen and Jerry E. Vaughan (eds.),*Handbook of set-theoretic topology*, North-Holland Publishing Co., Amsterdam, 1984. MR**776619****[9]**John R. Martin and Sam B. Nadler Jr.,*Examples and questions in the theory of fixed-point sets*, Canad. J. Math.**31**(1979), no. 5, 1017–1032. MR**546955**, https://doi.org/10.4153/CJM-1979-094-5**[10]**John R. Martin and E. D. Tymchatyn,*Fixed point sets of 1-dimensional Peano continua*, Pacific J. Math.**89**(1980), no. 1, 147–149. MR**596925****[11]**John R. Martin, Lex G. Oversteegen, and E. D. Tymchatyn,*Fixed point set of products and cones*, Pacific J. Math.**101**(1982), no. 1, 133–139. MR**671845****[12]**Helga Schirmer,*Fixed point sets of continuous self-maps*, Lecture Notes in Math., vol. 886, Springer, Berlin-New York, 1981, pp. 417–428. MR**643019****[13]**F. D. Tall,*Set-theoretic consistency results and topological theorems concerning the normal Moore space conjecture and related problems*, Ph.D. Thesis, University of Wisconsin, Madison, 1969.**[14]**L. E. Ward Jr.,*Fixed point sets*, Pacific J. Math.**47**(1973), 553–565. MR**0367963**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
54H25,
03E35,
54A35

Retrieve articles in all journals with MSC: 54H25, 03E35, 54A35

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1984-0742428-1

Keywords:
Fixed point set,
complete invariance property,
(Cartesian) product space,
topological group,
continuum hypothesis,
Martin's Axiom

Article copyright:
© Copyright 1984
American Mathematical Society