## 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

### MSC:

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