Compactness and Compactification from the statistical point of view.

The concept of statistical convergence was conceived in the mind of H. Fast, which is a generalization of usual convergence of sequence from statistical point of view based on natural density of subsets of natural numbers. Then a number of mathematicians like A. Caserta, L.D.R. Kočinac, J. Connor, J. Kline, G. Di Maio, J.A. Fridy, H.I. Miller, M.K. Khan, C. Orhan, P. Das etc. extended this research area by passing one of the important properties namely smallness or compactness properties of topological spaces; actually the way for looking into subsequence could not provide anything new in this regard. A way of thinking about convergence of subsequence is found and seen that it is consistent with the usual one that enable to establish a notion of compactness which is not even equivalent to the usual compactness of metric spaces. Blended change have been seen while developing the theory of statistical compactness. A realization of statistical compactness as an application of Cantor-Bendixson derivative is presented in Euclidean spaces and subsequently extended to the product space .

The notions of I-compactness and I*-compactness are introduced via ideal I on N and I-nonthin subsequences, and their continuous image, hereditary property, productive property are discussed. Corelations between the said compactness and their relation with usual compactness are exhibited under some conditions namely shrinking conditions on I. The discussion is continued towards I-US spaces and I-sequential spaces. Various properties I-continuous functions and I-proper maps are displayed. One point I-compactification of a topological space is revealed via the radiant of an ideal on N. A class of ideals has been identified for which one point I-compactification of a metrizable space coincides with usual one point compactification of that space.