Properties of Compact Set in G-metric Space

Abstract

In mathematical analysis, topology is built by metric spaces. A metric space is a set in which the metric axioms are satisfied. Many mathematicians discuss the generalization of metric spaces. One of them is the concept of G-metric space, denoted by (X, G), which was introduced in 2006. Within the metric space, there are many special sets that have played an important role in developments in the field of mathematical analysis, in particular compact sets. A set is said to be compact if each open cover of the set has finite subcovers. The properties of compact sets have been discussed in metric spaces, Hausdorff spaces, topological spaces, and fuzzy metric spaces. However, there are no researchers who discuss the properties of compact sets in G−metric spaces. Therefore, to expand the discussion of the concept of G−metric space and the properties of compact sets that apply to it, this article discusses the proof of theorems related to the properties of compact sets in G−metric space. Compact sets in G−metric spaces have the properties of being closed and bounded. However, not all closed and bounded sets are compact sets. To prove the compactness of a set in G−metric space, in addition to using the concept of open covers, it can also be proven by the G−completeness and G−totally boundedness properties of a set.