On the mathematical theory of transfinite dimensions and its application in physics

Following the mathematical theory of infinite dimensional spaces, we introduce basic definitions and theorems. We show that the three different fundamental notions of dimension coincide only for separable metric spaces. Subsequently, the degree of infinite dimensionality is considered, which leads us to the notion of transfinite dimensions and the introduction of stratification to a wide class of spaces. These particular spaces are the main subject of E-infinity theory of high energy physics.