Metric spaces related to Abelian groups


  • Amir Veisi Yasouj University
  • Ali Delbaznasab Shahid Bahonar University



G-metric space, L-group, Dedekind-complete group, densely ordered group, continuity


When working with a metric space, we are dealing with the additive group (R, +). Replacing (R, +) with an Abelian group (G, ∗), offers a new structure of a metric space. We call it a G-metric space and the induced topology is called the G-metric topology. In this paper, we are studying G-metric spaces based on L-groups (i.e., partially ordered groups which are lattices). Some results in G-metric spaces are obtained. The G-metric topology is defined which is further studied for its topological properties. We prove that if G is a densely ordered group or an infinite cyclic group, then every G-metric space is Hausdorff. It is shown that if G is a Dedekind-complete densely ordered group, (X, d) a G-metric space, A ⊆ X and d is bounded, then f : X → G with f(x) = d(x, A) := inf{d(x, a) : a ∈ A} is continuous and further x ∈ clXA if and only if f(x) = e (the identity element in G). Moreover, we show that if G is a densely ordered group and further a closed subset of R, K(X) is the family of nonempty compact subsets of X, e < g ∈ G and d is bounded, then d' (A, B) < g if and only if A ⊆ Nd(B, g) and B ⊆ Nd(A, g), where Nd(A, g) = {x ∈ X : d(x, A) < g}, dB(A) = sup{d(a, B) : a ∈ A} and d' (A, B) = sup{dA(B), dB(A)}.


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Author Biographies

Amir Veisi, Yasouj University

Faculty of Petroleum and Gas

Ali Delbaznasab, Shahid Bahonar University

Department of Mathematics


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How to Cite

A. Veisi and A. Delbaznasab, “Metric spaces related to Abelian groups”, Appl. Gen. Topol., vol. 22, no. 1, pp. 169–181, Apr. 2021.



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