Introduction

Abstract

This preliminary chapter introduces some suggestive ideas that lead to the versions of transfiniteness that we shall be examining in this book. The second section herein speculates about the possibility of connecting infinite graphs together at their infinite extremities to get a structure that is indescribable as a conventional graph. It points out that the conventional definitions of a graph lead to unwieldy constructions of the desired transfinite structure, one that is unnecessarily complicated. The third section then presents an alternative — and possibly novel — definition of a graph that is more amenable to transfinite generalization. The last section discusses a lacuna in the theory of infinite electrical networks that can be closed only by introducing connections to the network at infinity. This is the first step toward transfiniteness for electrical networks. But, first of all, we start by defining the symbols and phrases we will be using in this book.