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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
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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 }$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1982-0637180-8
PII: S 0002-9939(1982)0637180-8
Article copyright: © Copyright 1982 American Mathematical Society