O'Farrell, Anthony G. (2004) When uniformly-continuous implies bounded. Bulletin of the Irish Mathematical Society, 53 (Summer). pp. 53-56. ISSN 0791-5578

Full text not available from this repository.

Abstract

Let (X,p) and (Y,σ) be metric spaces. A function f : X → Y is (by definition) bounded if the image of f has finite σ-diameter. It is well-known that if X is compact then each continuous f : X → Y is bounded. Special circumstances may conspire to force all continuous f : X → Y to be bounded, without Y being compact. For instance, if Y is bounded, then that is enough. It is also enough that X beconnected and that each connected component of Y be bounded. But if we ask that all continuous functions f : X → Y , for arbitrary Y, be bounded, then this requires that X be compact. What about uniformly-continuous maps? Which X have the property that each uniformly-continuous map from X into any other metric space must be bounded?

Item Type:	Article
Keywords:	Uniformly-continuous; Bounded; f : X → Y; Epsilon-step territories.
Academic Unit:	Faculty of Science and Engineering > Mathematics and Statistics
Item ID:	1812
Depositing User:	Prof. Anthony O'Farrell
Date Deposited:	26 Jan 2010 12:46
Journal or Publication Title:	Bulletin of the Irish Mathematical Society
Publisher:	Irish Mathematical Society
Refereed:	No
URI:	Publisher
Use Licence:	This item is available under a Creative Commons Attribution Non Commercial Share Alike Licence (CC BY-NC-SA). Details of this licence are available here

Repository Staff Only(login required)

Item control page

Downloads

Origin of downloads