O'Farrell, Anthony G. (2004) When uniformly-continuous implies bounded. Bulletin of the Irish Mathematical Society, 53 (Summer). pp. 53-56. ISSN 0791-5578Full text not available from this repository.
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?
|Keywords:||Uniformly-continuous; Bounded; f : X → Y; Epsilon-step territories.|
|Subjects:||Science & Engineering > Mathematics & Statistics|
|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|
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