Exploring topological properties using ideal structure.

In the realm of general topology, ideal has opened a new horizon. The year 2000 saw the raise of ideal convergence which was a generalization of usual convergence. Ideal convergence sheds light on limit points of functions, closure operator, continuity, compactness etc. The notions of -Fréchet compactness and -Fréchet compactness are introduced via ideals , of subsets of a non-empty set and -nonthin function. It has been seen that, in first countable spaces Fréchet compactness and -Fréchet compactness are equivalent. A family of ideals has been established for which -Fréchet compactness coincides with -Fréchet compactness. The concepts of -functionally compactness and -functionally compactness are introduced in the context of ideals , of subsets of a non-empty set and -nonthin function and showed that these are different from compactness even in metric spaces. The relation between the aforementioned compactness with usual compactness are explored by using the class of ideals.