This book presents, in a clear and structured way, the set function mathcal{T} and how it evolved since its inception by Professor F.Burton Jones in the 1940s. It starts with a very solid introductory chapter, with all the prerequisite material for navigating through the rest of the book.It then gradually advances towards the main properties, Decomposition theorems, mathcal{T}-closed sets, continuity and images, to modern applications. The set function mathcal{T} has been used by many mathematicians as a tool to prove results about the semigroup structure of the continua, and about the existence of a metric continuum that cannot be mapped onto its cone or to characterize spheres.Nowadays, it has been used by topologists worldwide to investigate open problems in continuum theory. This book can be of interest to both advanced undergraduate and graduate students, and to experienced researchers as well.Its well-defined structure make this book suitable not only for self-study but also as support material to seminars on the subject.Its many open problems can potentially encourage mathematicians to contribute with further advancements in the field.