The Set Function $${\mathcal {T}}$$

Abstract

We prove basic results about the set function \({\mathcal {T}}\) defined by F. Burton Jones (Amer J Math 70:403–413, 1948) to study the properties of metric continua. We define this function on compacta, and then we concentrate on continua. In particular, we present some of the well known properties (such as connectedness im kleinen, local connectedness, semi-local connectedness, etc.) using the set function \({\mathcal {T}}\). The notion of aposyndesis was the main motivation of Jones to define this function. We present some properties of a continuum when it is \({\mathcal {T}}\)-symmetric and \({\mathcal {T}}\)-additive. We give properties of continuum on which \({\mathcal {T}}\) is idempotent, idempotent on closed sets and idempotent on contina. We also present results about the set functions \({\mathcal {T}}^n\), when \(n\in {\mathbb {N}}\).