TY - JOUR
AU - Veisi, Amir
AU - Delbaznasab, Ali
PY - 2021/04/01
Y2 - 2024/02/27
TI - Metric spaces related to Abelian groups
JF - Applied General Topology
JA - Appl. Gen. Topol.
VL - 22
IS - 1
SE -
DO - 10.4995/agt.2021.14446
UR - https://polipapers.upv.es/index.php/AGT/article/view/14446
SP - 169-181
AB - <p>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 cl<sub>X</sub>A 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† N<sub>d</sub>(B, g) and B âŠ† N<sub>d</sub>(A, g), where N<sub>d</sub>(A, g) = {x âˆˆ X : d(x, A) < g}, d<sub>B</sub>(A) = sup{d(a, B) : a âˆˆ A} and d' (A, B) = sup{d<sub>A</sub>(B), d<sub>B</sub>(A)}.</p>
ER -