In this paper, a local approach to the concept of Hudetz $g$-entropy is presented. The introduced concept is stated in terms of Hudetz $g$-entropy. This representation is based on the concept of $g$-ergodic decomposition which is a result of the Choquet's representation Theorem for compact convex metrizable subsets of locally convex spaces.

In this note, the concepts of (G1,G2)-open, (G1,G2)-closed, (G1,G2)-quotient and (G1,G2)-perfect mappings on arbitrary sets are introduced some theorems them established firstly. particular, results improve corresponding in [17]. Secondly, we give a partial answer to question posed by L. Liu [14]. Finally, properties G-topological groups, G-connectedness totally G-disconnectedness groups discus...