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
MathSciNet review: 637180
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 .
-  Frank B. Miles, Compact, totally disconnected sets that contain 𝐾-sets, Michigan Math. J. 21 (1974), 315–319 (1975). MR 0405000
-  R. Kaufman, A functional method for linear sets, Israel J. Math. 5 (1967), 185–187. MR 0236607, https://doi.org/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 0313721
-  Georg Cantor, Ueber unendliche, lineare Punktmannichfaltigkeiten, Math. Ann. 17 (1880), no. 3, 355–358 (German). MR 1510071, https://doi.org/10.1007/BF01446232
-  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
- F. B. Miles, Compact, totally disconnected sets that contain -sets, Michigan Math. J. 21 (1974), 315-319. MR 0405000 (53:8796)
- R. Kaufman, A functional method for linear sets, Israel J. Math. 5 (1967), 185-187. MR 0236607 (38:4902)
- Y. Katznelson, An introduction to harmonic analysis, Wiley, New York, 1968, pp. 184-185. MR 0248482 (40:1734)
- F. B. Miles, Existence of special -sets in certain locally compact abelian groups, Pacific J. Math. 44 (1973), 219-232. MR 0313721 (47:2275)
- G. Cantor, Ueber enendliche, lineare Punktmannichfaltigkeiten, Math. Ann. 17 (1880), 355-358. MR 1510071
- K. Kuratowski, Topology, Vol. I, Academic Press, New York, 1966. MR 0217751 (36:840)