Topics in Exponential Spaces

Theory of exponential spaces, commonly known as hyperspace, was a very fascinating subject in Pure Mathematics in the last century. A number of mathematical geniuses was involved in the development of hyperspace theory; Hausdorff, Kuratowski, Michael, Kelley, Fell were among them, to name a few. In our sense any type of collection of subsets with some special features having a suitable topology has been designated as hyperspace. The present treatise is a colorful manifestation of various aspects of hyperspace theory. This short work has been developed for the study of hyperspaces containing H-continua, N-continua etc. as their constituent elements and finally topologisation of the collection suitably. Inspired by a special hyperspace, a new topological algebraic structure, named as "topological quasi-vector space", has been introduced and studied with meticulous detail. ''Ideal'' of this structure has been defined and its several properties are studied. Lastly, theory of hyperspace has been utilized to study multi-valued functions. This short course is useful for those who are interested in topological algebraic structure and are doing research in various facets of it.