A NOTE ON METRIC PRESERVING FUNCTIONS

The purpose of this note is to study modifications of the Euclidean metric on R with the following property: There is a monotone sequence of closed balls with empty intersection.


Introduction
Definition 1.We call a function f : R + → R + metric preserving iff f (d) : M × M → R + is a metric for every metric d : M ×M → R + , where (M, d) is an arbitrary metric space and R + denotes the set of nonnegative reals.We denote by M the set of all metric preserving functions.(See Borsík [1], Borsík [2], Terpe [3].) The following result is well known (see Borsík [1], Terpe [3]).
Proposition 1.If f : R + → R + is a concave function vanishing exactly at orgin than it is metric preserving.
It is well known that there is a complete metric space with the following property: There is a monotone sequence of closed balls with empty intersection. ( In Jůza [4] such a metric space (which is not discrete) has been constructed by a modification of the Euclidean metric on R, where R denotes the set of reals.
For each f ∈ M denote by d f the metric on R defined as follows We call d f a modification of the Euclidean metric on R. (See Terpe [3].) Example 1. Define f : R + → R + as follows: In Jůza [4] it is shown that f ∈ M and the metric space (R, d f ) has the property (1.1).The proof of (1.1) is based on the following property of the metric space (R, d f ): For each compact set K there is a closed ball S and there is

Main Results
Theorem 1.Let f ∈ M. Suppose that there are g, h : R + → R + such that g, h are nonincreasing, and they are not constant in each neighborhood of the point +∞, (2.1) g(x) ≤ f (x) ≤ h(x) in some neighborhood of the point +∞, (2.2) lim x→+∞ g(x) = lim x→+∞ h(x).
Let S be a closed ball with the centre s and the radius δ.
Example 2. Define f : R + → R + as follows: It is not difficult to verify that f ∈ M and the metric space (R, d f ) has the property (1.2) (which yields also the property (1.1) ), however f is not monotone on every neighborhood of the point +∞.

Example 3. Define
, if x ∈ (3n − 2, 3n + 1](n = 1, 2, 3, . . .).It is not difficult to verify that f ∈ M and (R, d f ) is a metric space with the property (1.1), which has not the property (1.2).Indeed, the intersection of the sequence of closed balls {S n } ∞ n=1 (where S n has the centre x n = 3 • (2 n−1 − 1) and the radius ε n = 1 2 + 1 2 n +1 ) is empty.A characterization of metric preserving functions f such that the space (R, d f ) has the property (1.1) remains an open question.
.2) 1991 Mathematics Subject Classification.26A30.Key words and phrases.Metric preserving functions.The first named author wishes to express his appreciation to the Department of Mathematics of Youngstown State University for their hospitality during his stay.The second named author wishes to acknowledge a support from Youngstown State University Research Council.