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On the completeness of metric spaces with a \(w\)-distance
  • Aleksandar Kostic
Aleksandar Kostic
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Abstract

Abstract.  A \(w\)-distance on a metric space \(\left(X,d\right)\) is a function \(p:X\times X\to\left[0,\infty\right)\)  which is lower semicontinuous with respect to the second varibale, satisfies the triangle inequality and for all \(\varepsilon>0\) there exists a \(\delta>0\) such that \(p\left(z,x\right)\le\delta\) and \(p\left(z,y\right)\le\delta\) imply \(d\left(x,y\right)\le\varepsilon\) for all \(x,y,z\in X\). In this short note we prove that a metric space with a \(w\)-distance \(p\) is complete if and only if every sequence \(\ \left\{x_i\right\}\ \) such that \(\sum_{_{i=1}}^{\infty}p\left(x_i,x_{i+1}\right)<\infty\) converges.