Mappings that preserve Cauchy sequences. (English) Zbl 0657.54024

Given metric spaces X and Y, denote by C(X,Y), U(X,Y) and F(X,Y) the set of all continuous, uniformly continuous and Cauchy-sequence-preserving maps from X to Y, respectively. It is known that U(X,Y)\(\subseteq F(X,Y)\subset C(X,Y)\) and the author studies the question, when equality holds. For the following assume that Y is a non-trivial normed space: Then \(C(X,Y)=F(X,Y)\) if and only if X is complete and \(F(X,Y)=U(X,Y)\) if and only if X fails to have the property (V): A metric space has the property (V) if there is a sequence S without Cauchy-subsequence such that for all \(c>0\) there are k, n with \(0<d(S(k),S(n))<c.\)
If X is not complete then F(X,Y) is nowhere dense in C(X,Y) and if X has (V) then U(X,Y) is nowhere dense in F(X,Y), where C(X,Y) and F(X,Y) carry the topology of uniform convergence.
Reviewer: H.-P.Butzmann


54E35 Metric spaces, metrizability
54C05 Continuous maps
54E40 Special maps on metric spaces
54C35 Function spaces in general topology