A characterization of the uniform closure of the set of homeomorphisms of a compact totally disconnected metric space into itself
HTML articles powered by AMS MathViewer
- by Frank B. Miles PDF
- Proc. Amer. Math. Soc. 84 (1982), 264-266 Request permission
Abstract:
The limit index $\lambda (x)$ of a point $x$ in a compact metric space is defined. (Roughly: Isolated points have index 0, limit points have index 1, limit points of limit points have index 2, and so forth.) Then the following theorem is proved. Theorem 1. Let $E$ be a compact, totally disconnected metric space. Then the uniform closure of the set of homeomorphisms of $E$ into itself is the set ${C_\lambda }$ of continuous functions $f$ from $E$ to $E$ satisfying (1) $\lambda (x) \leqslant \lambda (f(x))\;for\;all\;x \in E$, and (2) if $y$ is not a condensation point of $E$, then ${f^{ - 1}}(y)$ contains at most one $x$ such that $\lambda (x) = \lambda (y)$. Further, the set of homeomorphisms of $E$ into $E$ is a dense ${G_\delta }$ subset of the complete metric space ${C_\lambda }$.References
- Frank B. Miles, Compact, totally disconnected sets that contain $K$-sets, Michigan Math. J. 21 (1974), 315–319 (1975). MR 405000
- R. Kaufman, A functional method for linear sets, Israel J. Math. 5 (1967), 185–187. MR 236607, DOI 10.1007/BF02771106
- Yitzhak Katznelson, An introduction to harmonic analysis, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0248482
- Frank B. Miles, Existence of special $K$-sets in certain locally compact abelian groups, Pacific J. Math. 44 (1973), 219–232. MR 313721
- Georg Cantor, Ueber unendliche, lineare Punktmannichfaltigkeiten, Math. Ann. 17 (1880), no. 3, 355–358 (German). MR 1510071, DOI 10.1007/BF01446232
- K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 84 (1982), 264-266
- MSC: Primary 54C40; Secondary 54E50
- DOI: https://doi.org/10.1090/S0002-9939-1982-0637180-8
- MathSciNet review: 637180