# Metric spaces related to Abelian groups

## Authors

• Amir Veisi Yasouj University
• Ali Delbaznasab Shahid Bahonar University

## Keywords:

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

## Abstract

When working with a metric space, we are dealing with the additive group (R, +). Replacing (R, +) with an Abelian group (G, âˆ—), 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 âŠ† X and d is bounded, then f : X â†’ G with f(x) = d(x, A) := inf{d(x, a) : a âˆˆ A} is continuous and further x âˆˆ 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 âˆˆ G and d is bounded, then d' (A, B) < g if and only if A âŠ† Nd(B, g) and B âŠ† Nd(A, g), where Nd(A, g) = {x âˆˆ X : d(x, A) < g}, dB(A) = sup{d(a, B) : a âˆˆ A} and d' (A, B) = sup{dA(B), dB(A)}.

## Author Biographies

### Amir Veisi, Yasouj University

Faculty of Petroleum and Gas

### Ali Delbaznasab, Shahid Bahonar University

Department of Mathematics

## References

A. Arhangel'skii and M. Tkachenko, Topological groups and related structures, Atlantis press/ World Scientific, Amsterdam-Paris, 2008. https://doi.org/10.2991/978-94-91216-35-0

R. Engelking, General topology, Sigma Ser. Pure Math., Vol. 6, Heldermann Verlag, Berlin, 1989.

L. Gillman and M. Jerison, Rings of Continuous Functions, Springer-Verlag, Berlin/Heidelberg/New York, 1976.

I. Kaplansky, Infinite Abelian Groups, University of Michigan Press, 1954.

O. A. S. Karamzadeh, M. Namdari and S. Soltanpour, On the locally functionally countable subalgebra of C(X), Appl. Gen. Topol. 16, no. 2 (2015), 183-207. https://doi.org/10.4995/agt.2015.3445

M. Namdari and A. Veisi, Rings of quotients of the subalgebra of C(X) consisting of functions with countable image, Inter. Math. Forum 7 (2012), 561-571.

D. J. S. Robinson, A course in the theory of groups, second edition, Springer-Verlag New York, Inc. 1996.

J. Rotman, An Introduction to the Theory of Groups, Vol. 148, 4th edition Springer, New York, 1995.

A. Veisi, The subalgebras of the functionally countable subalgebra of C(X), Far East J. Math. Sci. (FJMS) 101, no. 10 (2017), 2285-2297. https://doi.org/10.17654/MS101102285

A. Veisi, Invariant norms on the subalgebras of \$C(X)\$ consisting of bounded functions with countable image, JP Journal of Geometry and Topology 21, no. 3 (2018), 167-179. https://doi.org/10.17654/GT021030167

A. Veisi, ec-Filters and ec-ideals in the functionally countable subalgebra of C*(X), Appl. Gen. Topol. 20, no. 2 (2019), 395-405. https://doi.org/10.4995/agt.2019.11524

A. Veisi and A. Delbaznasab, New structure of norms on Rn and their relations with the curvature of the plane curves, Ratio Mathematica 39 (2020), 55-67.

S. Willard, General Topology, Addison-Wesley, 1970.

2021-04-01

## 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.

## Section

Regular Articles  