Metric-preserving Functions, W-distances and Cauchy W-distances

T. Khemaratchatakumthorn, I. Termwuttipong

Abstract

A function $f : [0,\infty) \rightarrow [0,\infty)$ is called metric-preserving if for every metric space $(X,d), f \circ d$ is a metric on X. The notation of w-distance on a metric space was introduced by Shioji, Suzuki and Takahashi in 1998. By a w-distance on a metric space (X,d), they mean a function $p : X \times X \rightarrow [0,\infty)$ having the properties that for all $x,y,z \in X, p(x,z) \leq p(x,y)+p(y,z), p(x,\cdot)$ is lower semicontinuous and for any $\varepsilon > 0$, there is a $\delta > 0$ such that $p(z,x) \leq \delta$ and $p(z,y) \leq \delta$ imply $d(x,y) \leq \varepsilon$. Then we call such a p a Cauchy w-distance if every Cauchy sequence ($x_n$) in (X, d) has the property relating to p that for every $\varepsilon > 0$, there exists a positive integer N such that $p(x_n, x_m) < \varepsilon$ for all $m > n \geq N$. Our purpose is to show that the metric $f \circ d$ is a w-distance on (X, d) if f is lower semicontinuous and it is a Cauchy w-distance if f is continuous.