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When uniformly-continuous implies bounded

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

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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

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