Hostname: page-component-848d4c4894-89wxm Total loading time: 0 Render date: 2024-07-07T17:00:23.199Z Has data issue: false hasContentIssue false

Existence of Certain Analytic Homeomorphisms

Published online by Cambridge University Press:  20 November 2018

Z. A. Melzak*
Affiliation:
McGill University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This note has its origin in the following problem: do there exist non-trivial increasing continuous functions on [0, l] to [0, l], which map the following sets in [0, l] onto themselves: the rational, the algebraic and the transcendental numbers? One such function is obviously f(x) = x; more generally, f(x) = (c + l)x/(cx + 1), with c rational and non-negative, satisfies the conditions. Let G denote the space of order-preserving homeomorphisms of [0, l} onto [0, l], in the uniform metric. It follows from Theorem 1 below that the set S of all such functions i s dense in G. S is clearly a subgroup of G and one may ask what a r e its group-theoretic properties. We shall not consider these questions.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1959

References

1. Franklin, P., Analytic transformations of linear everywhere dense point sets, Trans. Amer. Math. Soc. 27(1925), 91-100.Google Scholar