On \(\beta\)-\(\gamma\)-connectedness and \(\beta_{(\gamma,\delta)}\)-continuous functions

The purpose of this work is to present the idea of \(\beta-\gamma\)-separated sets, examine their characteristics in topological spaces and then define the notation for \(\beta-\gamma\)-connected and \(\beta-\gamma\)-disconnectedness.\ In addition, the study of topological qualities that involves for \(\beta-\gamma\)-connected spaces via \(\beta-\gamma\)-separated sets.\ An analysis is conducted on the properties of \(\beta-\gamma\)-connected spaces and how they behave under \(\beta_{(\gamma,\delta)}\)-continuous functions.\ We also provide the ideas of \(\beta-\gamma\)-components in a space \(X\) and \(\beta-\gamma\)-locally connected spaces.