A characterization of the uniform closure of the set of homeomorphisms of a compact totally disconnected metric space into itself

Author:
Frank B. Miles

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

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Abstract: The limit index of a point 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* *be a compact, totally disconnected metric space. Then the uniform closure of the set of homeomorphisms of* *into itself is the set* *of continuous functions* *from* *to* *satisfying*

(1) , and

(2) *if* *is not a condensation point of* , *then* *contains at most one* *such that* .

*Further, the set of homeomorphisms of* *into* *is a dense* *subset of the complete metric space* .

**[1]**Frank B. Miles,*Compact, totally disconnected sets that contain 𝐾-sets*, Michigan Math. J.**21**(1974), 315–319 (1975). MR**0405000****[2]**R. Kaufman,*A functional method for linear sets*, Israel J. Math.**5**(1967), 185–187. MR**0236607**, https://doi.org/10.1007/BF02771106**[3]**Yitzhak Katznelson,*An introduction to harmonic analysis*, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR**0248482****[4]**Frank B. Miles,*Existence of special 𝐾-sets in certain locally compact abelian groups*, Pacific J. Math.**44**(1973), 219–232. MR**0313721****[5]**Georg Cantor,*Ueber unendliche, lineare Punktmannichfaltigkeiten*, Math. Ann.**17**(1880), no. 3, 355–358 (German). MR**1510071**, https://doi.org/10.1007/BF01446232**[6]**K. Kuratowski,*Topology. Vol. I*, New edition, revised and augmented. Translated from the French by J. Jaworowski, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, Warsaw, 1966. MR**0217751**

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DOI:
https://doi.org/10.1090/S0002-9939-1982-0637180-8

Article copyright:
© Copyright 1982
American Mathematical Society