2 - Mathematical Prerequisites

Abstract

This chapter describes different aspects of mathematical prerequisites. A subset of a base from which the base can be constructed is itself a base and, hence, is termed a sub-base. It is assumed that z is a point in an open set and is contained in an interval entirely in the open set. One can construct a new interval whose end points have rational coordinates, that is entirely contained in the old interval, and that contains z. All open sets are the union of intervals with rational coordinates for the end points. When the base is such that its members can be counted out according to the positive integers so that every set in the base is counted in the counting process, then the base is said to be countable and the topological space is said to be completely separable. For example, the rational numbers are countable, so points with rational coordinates are equivalent to pairs of rational numbers and are hence countable, and finally, intervals with rational end points are countable.